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This essay is an admirable start! I think we can expand on the categories quite a bit and provide more examples. As the author himself says:

'I would encourage an effort to build on this, so that a more complete resource would be available to Chess Variant designers or problemists.'

I think there may be an obvious category which is 'Imitator' as in the Chameleon / Mimotaur. [Edited addition: I see now that David has written an entirely separate and also excellent essay on imitators.]

The 'Ooze' has a cousin in the Amoeba from Hedden's Microorganism Chess which I think introduces some more original categories of pieces as well...

Movement: define a 2-stepper PIECE to make 1 or 2 moves like that PIECE. The direction of movement can change with every move, unlike the case of a rider.

I use the name 'Commoner' for the usual (Wazir+Ferz) compound. The Lion has all the moves of a 2-stepper Commoner, with no restrictions. It can make a null move by returning to the square it started on. It can capture by igui. It can even make two captures in one turn. The Lion also can make a single Squirrel leap. In Jetan the Chief is a 3-stepper Commoner, which always makes exactly 3 steps, plus some other restrictions.

My recent games use the name 'War Elephant' for the 2-stepper Ferz, which has also been called Jester and Free Padwar. None of these chess variants allow the piece to make a null move. Again, the original definition in Jetan was most likely a Chained Padwar, moving exactly two steps.

I have a vague memory (several years old) of a 2-stepper Knight in a some chess variant. It may have been allowed to capture by igui. Not to be confused with the Ubi-Ubi. EDIT: The SUPER KNIGHT '... can capture a maximum of two enemy pieces in one move.' in Lim Ther Peng's Supremo Superchess.

The Bent Hero and Bent Shaman in Joe Joyce's Lemurian Shatranj are 'Inclusive Compound' pieces. They can be regarded as a '2-steppers' where the only 2-step moves allowed are those of two different pieces.

This is a really well done explanation of the various type of pieces that have been invented so far. While missing out on a few advanced concepts (mostly from large Shogi variants), it covers all the common (and much of the uncommon) variables very well.

Under movement, a stepper and multiple steppers deserve some mention. Single step pieces, like the King, Commoner, various Shogi Generals, Fers, Wazir, etc. are really a class all thier own. While they can be interpreted as either short range riders or leapers, they aren't really either.

Multi steppers, such as the Chu Shogi Lion make several moves as single steps. Other multi steppers are a sort of limited range rider. They can only stop at thier far square, but must pass through the other squares on the way. The Xiang Qi Horse is a bent piece of this sort.

Some mention of the special subcase of rider, the slider, should be made. A slider moves by multible steps (like a Rook, Bishop, Queen, or Lance). A non-sliding rider moves by multiple leaps (like the Knightrider and other oblique riders).

Under capture, you should mention that some peices (again from obscure Shogi variants) can capture friendly peices as well.

Some mention of restricted movement zones could be useful as well. The most well known examples are from Xiang Qi, where the Elephant cannot cross the river and the General and Advisors cannot leave the palace.

A related concept from Janggi (or Changgi), Korean chess, is enhanced movement zones. Various pieces have additional powers of movement when in the palace.

I'm sure there are some other things that are missing as well, but that's what I've noticed at this time.

Piece can also be classified byit's royality:
Non-royal;
Royal, can be further divided into these groups:
1. If all pieces of that kind are captured, it's owner lloses;
2. If one of pieces of that kind is checkmated (or stalemated, depending on game's rules), it's owner loses.

I was thinking of doing something along the lines of this page when I came across it on an early CD-ROM of this site. One detail I found surprising is the claim that "compound" was coined by John William Brown in Meta-Chess. Checking out earlier sources, I found V. R. Parton using the word. So I have removed the claim that it was coined in Meta-Chess.

I mentioned in my earlier comment that I already had something like this page in mind. What I actually had in mind was a taxonomic index of the pieces in the Piececlopedia, which would be helpful for getting a better idea of what is missing and could be added. I have made an initial draft of it at /piececlopedia.dir/pindex.html, which I plan to eventually make into the new index page for the Piececlopedia directory. I have sometimes used the terminology from this page, but I have also added new terminology for types of pieces not covered here. Please take a look and let me know if anything is wrongly classified or might be better classified.

With alternative terminologies (probably not a good idea):
# Chess-based Pieces
## Chess Pieces
Bishop
King
Knight
Pawn
Queen
Rook
## Hybrids
Amazon - Queen + Knight.
Archbishop / Cardinal - Bishop + Knight
Chancellor / Marshall / Empress - Rook + Knight
Crowned Rook / Dragon King - King + Rook
Crowned Bishop / Dragon Horse - King + Bishop
## Compounds
Crowned Knight / Centaur / Princess - King + Knight
Dragon - Pawn + Knight
# Leapers
## Single-Pattern
Alfil / Elephant (2-2 leaper)
Antelope (3-4 leaper)
Camel (1-3 leaper)
Dabbabah (0-2 leaper)
Ferz (1-1 leaper)
Flamingo (1-6 leaper)
Giraffe (1-4 leaper)
Knight (1-2 leaper)
Wazir (0-1 leaper)
Zebra (2-3 leaper)
## Double-Pattern
Alibaba - Alfil + Dabbabah
Bison - Camel + Zebra
Carpenter - Knight + Dabbabah
Gnu / Wildebeest - Knight + Camel
Kangaroo (Newton) - Knight + Alfil
Man / Commoner - Ferz + Wazir
Okapi Knight + Zebra
Phoenix / Waffle - Wazir + Alfil
Root 50 Leaper - 5-5 + 1-7 leaper
Wizard - Camel + Ferz
## Triple-Pattern
Buffalo - Knight + Camel + Zebra
Centaur - Man (Ferz + Wazir) + Knight
Champion - Wazir + Dabbabah + Alfil
FAD - Ferz + Alfil + Dabbabah
Squirrel - Alfil + Dabbabah + Knight
## Asymmetric
Barc - Reverse of Crab. Leaps as Knight narrow backwards or wide forwards.
Crab - Leaps as Knight narrow forwards or wide backwards.
Gold General - As a Wazir or forward one space
Honorable Horse / Shogi Knight - As Knight but only narrow forwards
Mushroom - Forward Knight, narrow backwards Knight, or backwards Ferz
Silver General - As a Ferz or forward one space
Fibnif - Ferz + narrow Knight
## Darters
Mao / Chinese Chess Knight - Moves one space orthogonally, followed by one space diagonally outward
Moa - Moves one space diagonally, followed by one space orthogonally outward
## Double
Single-Step Rhino. Mao + Wazir. By Peter Aronson.
## Snipers
Lion (Murray) - Moves as Alfil or Dabbabah, or captures as Ferz or Wazir
# Riders
## Simple
Bishop (1-1 rider)
Nightrider (1-2 rider)
Rook (0-1 rider)
Zebrarider (2-3 rider)
## Asymmetric
Lance - Moves as Rook straight forward
Zag-Zag - Moves vertically or along ne-sw diagonal.
Zag-Zig - Moves vertically or along nw-se diagonal.
Zig-Zag. Piece moves horizontally or along ne-sw diagonal.
Zig-Zig. Piece moves horizontally or along nw-se diagonal.
## Reflectors
Archbishop - Moves as Bishop but can make reflection at side of board
Bishop, Reflecting - Moves as a Bishop, but can also reflect off of multiple sides of the board.
## Multipath
Cavalier - RennChess piece that steps one diagonally then slides orthogonally, or steps one orthogonally then slides diagonally
Duke - Piece from RennChess that steps one orthogonally then slides diagonally, or slides diagonally then steps one orthogonally.
Ubi-Ubi - Can make arbitrary many knightmoves in one turn.
Sissa - Move exists of moving a number of squares as rook and an equal number of squares as bishop..
Crooked Queen - Rook + Crooked Bishop
## Compounds
Raven - Rook + Nightrider.
Unicorn (2) - Bishop + Nightrider.
## Rider + Leaper Hybrids
BD or Bede - Bishop + Dabbbabah
Caliph - Bishop + Camel
Canvasser - Rook + Camel
## Leaper
Griffon - Historic piece that steps one space diagonally then slides like a Rook.
Hippogriff - Limited version of the Griffon that must slide at least three squares.
Manticore - Moves one space orthogonally, then slides outward as a Bishop.
## Doubly Bent
Fox - Moving in a Wazir-Bishop-Wazir pattern.
Rabbit - Moving in a Knight-Nightrider-Knight pattern.
Wolf - Moving in a Ferz-Rook-Ferz pattern.
## Double
Crooked Bishop - Moves in a diagonal zigzagline.
## Curved
Rose - Can make consecutive knightmoves in a circle.
## Constricted
Edgehog - Moves as a Queen, but only to or from an edge.
Soucie - Moves on queen-lines exactly as many squares as there are pieces on that line.
# Alternative Pieces
## Alternate Pawns
Berolina Pawn Travels diagonally and captures straight forward
Berolina Plus Pawn - Like Berolina but can also capture horizontally
Pawn (Xiangqi) - Moves one space straight forward
Sergeant - A combination of the Berolina and usual Pawns.
Steward - Omnidirectional Pawn.
Pincer Pawn - Travels as rook, and takes by enclosing.
## Alternate Royal
General (Xiang Qi) - Moves like Wazir, but with royal restrictions.
King Battler - King usually moves as a Queen.
Scorpion - King with grasshopper power.
# Hoppers
## Pure Hoppers
Bishopper - Moves along diagonal line to first square after jumped over piece.
Contragrasshopper - Moves as queen but must always jump first.
Equihopper - Jumps across a piece in any direction with the same distance before and after the hurdle.
Grasshopper - Moves along queenlines to first square after jumped over piece
Kangaroo - Moves on Queen lines to first square after second jumped over piece.
Lion - Fairy piece that moves on Queen lines but must hop over exactly one piece.
Mao-hopper - Moves as knight must must jump over occupied orthogonal square at first movement.
Non-stop Equihopper - Jumps across a piece in any direction with the same distance before and after the hurdle.
## Chinese
Cannon / Pao - Travels as Rook but hops to capture
Leo Cannon + Vao
Vao - Travels as Bishop but hops to capture
## Leaper / Hopper Hybrids
Pancake - A piece that moves like a non-royal King or a Nightrider-style cannon.
## Curved
Windmill - Moves around piece.
# Double-Move Pieces
Lion - Chu Shogi piece
# Non-Displacement Capture
Advancer. Travels like a Queen, but captures by approach.
Bowman - Travels as knight, and takes a piece that is an additional knightsmove in same direction away.
Coordinator. Takes in `coordination' with king.
Locust - Moves along Queen lines to empty space immediately after first enemy piece, capturing the piece it hops over.
Long Leaper - Travels as queen, but takes by leaping.
Pawn - Google en passant!
Pincer Pawn - Travels as rook, and takes by enclosing.
Pushmi-Pullyu - Travels like a Queen, and captures by approach and withdrawal.
Withdrawer - Takes by moving away.

Why would you make a distinction between hybrids and compounds? What is that distinction in your mind? Why refer to leaper compounds as double and triple pattern instead of compounds? Why would you call the Mao and Moa Darters? What does that term mean for you? Why list bent riders under the heading of Leaper? Was that a mistake?

The Murray Lion is not a sniper. It captures by displacement. While based on the Cannon from Chinese Chess, the Leo and the Vao are not of Chinese origin.

Reflectors, Multi-Path, and Doubly Bent might be useful categories.

Simple compound is for a piece with only one move type.
Leaps & Leaps (Man {W+F}, Donut {D+N}, etc)
Rides & Rides (Queen {R+B}, Raven {R+NN}, etc)
Hops & Hops (Grasshopper {gR+gB}, Lion {pR+pB}, etc)
Hybrid is for a piece capable of 2 different types of move.
Rides & Leaps (Princess {B+N}, Empress {R+N}, etc)
Leaps & Hops (Pancake {K+pNN}, Scorpion {K+gQ}, etc)
I just think splitting Leaper Compounds into Double-Pattern & Triple-Pattern is neat.
Darter is a term for a blockable leaper, e.g. Alfil-darter {nA}, Cavabal {afasWasafW}, Mao {afsW}, Moa {afsF}, etc.
Leaper-Rider is a type of piece that makes a leap & may continue riding, e.g. Grypon {1,1 leaps then 1,0 rides}, Aanca {1,0 leaps then 1,1 rides}, etc.
An X/Y-sniper travels like piece X & captures like piece Y, so Murray Lion is an Alibaba/Pasha Sniper.
Argentinian Pieces aren't of Argentinian origin either.

Also long ago I have imagined a category called Hopper-Rider (hops not over certain number of obstacles, but over many numbers in regularity; for example Chinese Cannon Rider moves by hopping over 0, 2, 4, i. e. even number of obstacles and captures over 1, 3, 5, i. e. odd number). Just note it.

It seems that 'darter' is a synonym for 'lame leaper'. George Duke used that, but I don't know where he got it from.

Btw while making my ChessBreakTimes (not now doing so due to several reasons) I have distinguished different types of compounds, because categorizing was needed to not ruin chess-based RPG. So I named them Mix (different types of movement; note steppers and leapers as different classes in my game) and Hybrid (compound of already existing pieces; note that not all pieces were included in system). For example, Modern Elephant wasn’t a hybrid because there was no Alfil due to its weakness as RPG personage while Ferz and Elephant had some potential at least among other weak pieces; mix also was Squirrel due to absence of Alibaba in the game. And pieces like Raven (fsWbFbAfD) were mix; in general, if hybrid is added to game in update but one of his parts is missing in personages’ list, this hybrid is mix. However, now I understand that mix category would be better called hybrid, latter should be renamed to compound, and you will probably learn well from my mistakes)

Simple compound is for a piece with only one move type.

Since I contrast simple pieces with compound pieces, I would not refer to simple compounds.

Hybrid is for a piece capable of 2 different types of move.

Instead of using a different word than compound, I will just distinguish the kind of compound I'm speaking of.

I just think splitting Leaper Compounds into Double-Pattern & Triple-Pattern is neat.

I think it is less confusing to use the same term when the meaning doesn't change. I have now broken "Leaper Compounds" into "Double Leaper Compounds" and "Triple Leaper Compounds."

Darter is a term for a blockable leaper

Lame leaper is the more established term, and I don't understand the significance of calling them darters.

Leaper-Rider is a type of piece that makes a leap & may continue riding

Ralph Betza has already given us the term bent rider, and I want to favor established terminology over the use of new terms.

An X/Y-sniper travels like piece X & captures like piece Y, so Murray Lion is an Alibaba/Pasha Sniper.

I was thinking of rifle pieces, since rifles are usually used for sniping. Since a Pasha is a compound of a Man and an Alibaba, I don't think it properly counts as an X/Y-sniper. The examples Anthony Dickins gives for Snipers in A Guide to Fairy Chess have no overlapping capturing and non-capturing moves. In this case, though, one is a subset of the other, and if you allow that, all divergent pieces may be counted as Snipers. Since divergent is the more common term, and the Murray Lion has overlapping capturing and non-capturing moves, I do not favor calling it a Sniper.

Argentinian Pieces aren't of Argentinian origin either.

Since I don't think either one of us have been using that as a category, it's not a point in favor of placing Leo and Vao in a Chinese piece category.

Like this?

'Double' and 'triple' are more precise specifications of 'compound'. So I would just say 'double leaper' and 'triple leaper'. As opposed to 'elementary leaper'.

'Hybrid' wouod be a useful designation for compounds of different move types, like leaper + rider or leaper + hopper.

According to Christian Poisson, (m1,n1 ; m2,n2) Double Leaper is a piece making a leap of coordinates (m1,n1). If that arrival square is empty, it can make a second leap of coordinates (m2,n2).

Single-Step Rhino {WafsW} is a (0,1 ; 1,1) Double Leaper.

Serpent {(afz)F} is a (1,1 ; 0,1) Double Rider.

'Double' and 'triple' are more precise specifications of 'compound'. So I would just say 'double leaper' and 'triple leaper'.

As pointed out, a double leaper could be a piece that leaps twice, and a triple leaper could be a piece that leaps thrice. Sticking with the word compound makes what I mean less ambiguous.

'Hybrid' wouod be a useful designation for compounds of different move types, like leaper + rider or leaper + hopper.

One of the main contexts in which the word compound comes up is in speaking of the other compounds you could have of Chess pieces besides the Queen. If you called the Queen a compound but referred to the Knight-Bishop and Knight-Rook compounds as hybrids, that would confuse matters. So I am not in favor of using the word hybrid for a type of compound. It might be better used to describe a piece that combines the moves of two different pieces in a hybridized manner, such as the Chinese Chess Knight, which first moves as a wazir, then as a ferz, or bent riders, which start with a leap in one direction, followed by riding in a different direction.

As an afterthought, I could go along with speaking of hybrid compounds, which is different than substituting the word hybrid for compound. After Leapers, Riders, and Hoppers, I could include a new section on Hybrids, which includes hybrid compounds and other kinds of hybridized pieces.

The most qualified wizards here:

1) H.G.Muller

2) Fergus Duniho

3) HaruN Y

Thank you all)!

:)!