Nichtschach

The name is Notchess translated into German. This follows on from my previous 6x6x6 variants Unionschach, Sachsenschach, and Grossraumschach. These are extrapolations from the name of the 5x5x5 Raumschach.

I have since added what I eventually realised was a family of very similar variants but with pieces returning to the board as a different kind of piece. These variants therefore now have the collective name Flipped-return Nichtschach. One variant on that page, Springernichtschach, uses the same piece types and their representation can be inherited here.

Setup

Levels u and z

Level v

Level w

Level x

Level y

Pieces

	Simple radial linepieces, 4 aside of each, are represented by Shogi's promoted paired pieces:
	The ROOK moves any distance through empty intermediate cells in the 6 orthogonal directions, each of which is the intersection of 2 2d planes and at right angles to the other plane. It is unbound. It is represented by the promoted Wing.
	The BISHOP moves any distance through empty intermediate cells in the 12 standard diagonal directions, each of which is a diagonal of a 2d plane and at 45° to the other 2 planes. Each Bishop is bound to half the board. It is represented by the promoted Silver.
	The UNICORN moves any distance through empty intermediate cells in the 8 nonstandard diagonal directions, commonly called triagonal, which are all at 35° to all three planes. Each Unicorn is bound to a quarter of the board, which is why there are four of them aside. It is represented by the promoted Helm.
	Compound radial linepieces, 2 aside of each, are represented by Shogi's Kings and promoted linepieces:
	The QUEEN is the compound of Rook and Bishop. It inherits the Rook's lack of binding. It is represented by Shogi's King.
	The DUCHESS is the compound of Rook and Unicorn. It inherits the Rook's lack of binding. It is represented by Shogi's promoted Rook.
	The GOVERNOR is the compound of Bishop and Unicorn. The components' different bindings make the piece unbound. It is represented by Shogi's promoted Bishop.
	Simple 2:m:n leapers, 4 aside of each, are represented by Shogi's unpromoted paired pieces:
	The KNIGHT makes any 2:1:0 leap. The zero coordinate indicates that it always moves within one of the three planes, and the single odd coordinate that it is unbound and colourswitching. It is represented by the unpromoted Helm.
	The SEXTON makes any 2:1:1 leap. The nonzero smallest coordinate indicates that it never moves within one of the three planes, the 2 odd coordinates that it has the Bishop's binding, and the largest coordinate being their sum that it can triangulate. The name puns on its Square of Leap Length being 6. It is represented by the unpromoted Silver.
	The NINJA makes any 2:2:1 leap. The nonzero smallest coordinate indicates that it never moves within one of the three planes, and the single odd coordinate that it is unbound and colourswitching. The name puns on its Square of Leap Length being 9. It is represented by the unpromoted Wing.
	Compound 2:m:n leapers, 2 aside of each, are represented by Shogi's Golds and unpromoted linepieces:
	The CHURCHWARDEN is the compound of Knight and Sexton. The name means a higher ecclesiastical lay rank than Sexton. It is represented by Shogi's unpromoted Bishop.
	The SAMURAI is the compound of Knight and Ninja. The name means a Japanese warrior of higher rank than a Ninja. It is represented by Shogi's unpromoted Rook.
	The OBERON is the compound of Ninja and Sexton. The name, that of a Shakespeare character, is used here to stand for OBliquE baRON, from the SOLLS of its components being in a ratio of 2:3 like the radial Baron of which the Governor is the long-range version. It is represented by the Gold.
	Front rank, represented by the Shogi unpromoted Point:
	The pieces covering the front ranks are STOCKBROKERS, an enhanced form of the Broker (which follows rules 1 and 4 and cannot make ND moves at all) but distinct from the Pawnbroker (which follows rules 1 and 4 but extends rule 3 to all SD moves). It moves one step along any of the 9 forward radials with the restriction that the move must be:

Rules

Pawnbrokers reaching the far rank must be promoted to another array piece.

Victory is by capturing all enemy... Rooks and Rook compounds, or Bishops and Bishop compounds, or Unicorns and Unicorn compounds, or Knights and Knight compounds, or Sextons and Sexton compounds, or Ninjas and Ninja compounds.