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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.