Three-dimensional chess

First plane of a possible starting position, showing 11 possible pieces

Pieces

There are eight likely equivalents to rook, knight, bishop and queen, which have octahedral symmetry.
There are two more likely pieces with pyritohedral and tetrahedral symmetry. (Each with two different orientations.)
The 3D chess variant examined here has 15 named pieces.

Pieces can be bound in two different ways:
Bishop and jester are bound to black or white fields. This shall be called parity.
Envoy and ward can reach only a quarter of all fields, which shall be called red, green, blue and yellow. The term color shall be used for them.
So each field has a parity and a color.   (It is not enough to extend the checkered pattern of the plane into space.)

parity-bound and color-bound
A bishop on a white field can reach only white fields.	A jester on a white field can reach only white fields.	An envoy on a yellow field can reach only yellow fields.

grid with parity and color
6×6×6 chess space	2×2×2 unit with all eight combinations of parity and color

Shapes

The major pieces are represented by polyhedra, whose faces are orthogonal to the direction of capture.   (The design for the pawns is similar.)
In the small images below, the face colors denote direction types, e.g. dark red for axis, blue for plane diagonal, and dark green for space diagonal.

basic
(0, 0, 1) cube rook	(0, 1, 1) rhombic dodecahedron bishop	(1, 1, 1) octahedron envoy

combinations
small rhombicuboctahedron queen	great rhombicuboctahedron king	cuboctahedron marshal

jumpers
(0, 1, 2) tetrakis hexahedron knight	(1, 1, 2) deltoidal icositetrahedron* jester	(1, 2, 2) triakis octahedron* archer

reductions
pyritohedra squire (half knight)		tetrahedra ward (half envoy)

pawns
The traditional pawn captures along plane diagonals. Another one could capture along space diagonals. One could also combine the two. To fill the second plane with cheap cover, one may add a weak pawn, that captures only forward.
front dog	plane diagonal pike	space diagonal lance	both diagonals colonel

* The shapes of jester and archer are not Catalan, but integral.

Moves

These images indicate the directions of capture as points in a grid.
(In the parity layout the piece is on a white field. In the color layout it is on a yellow field.)

basic
parity rook bishop envoy color rook bishop envoy

combinations
parity queen / king marshal color queen / king marshal

jumpers
parity knight jester archer color knight jester archer

reductions
parity squire ward color squire ward

pawns
parity dog pike lance colonel color dog pike lance colonel

Space

An 8×8×8 cube (with 512 fields) would probably be too big. A 6×6×6 cube (with 216 fields) or 6×6×8 cuboid (with 288 fields) seem reasonable.

Possible starting positions

Every piece in the first plane, except queen and king, should exist twice. That would be 20 pieces.
The arrangement below is one example how that could look. It is not yet tested.
If the black positions are mirrored, no green pieces of the same type can threaten each other (but octahedra and tetrahedra can).

white polyhedra

white pawns

black pawns

black polyhedra