Each side starts with ten pawns on the third rank, "elemental" pieces on the second rank, and "complex" pieces on the first rank. The elemental pieces are the Rook, Knight, Bishop, and Squire. The complex pieces are the Lionheart King, Queen, Archbishop, Marshall, Cavalry, Gryphon, Hippogriff, and Unicorn.
The Rook, Bishop, and Queen move as they do in standard chess. The Knight is enhanced: It moves as a standard chess knight but, if the square it jumps to is unoccupied, it has the option of sliding one more space diagonally away from its starting rank and file. This sliding move may be used to capture. The Squire (S) moves like a standard chess King, but is not royal: it is not prohibited from moving onto a threatened square, and can be captured like any other piece.
The complex pieces consist of each compound of two elemental pieces, and a "bent" or extended form of each elemental move. The compounds are the Queen (Rook + Bishop, of course), Archbishop (bishop + knight), Marshall (Rook + Knight), and Cavalry (Knight + Squire).
The Gryphon is taken from Grande Acedrex, a very old variant of medieval chess (Shatranj), where it is called a giraffe. Its move is one step diagonally followed by any number of orthogonal steps away from its starting rank or file. It is a "bent rook". The Hippogriff* is its counterpart, starting with a single orthogonal step and proceeding diagonally away from its starting rank and file. It is a "bent bishop".
The Unicorn is a bit odd: it slides any number of spaces, alternating orthogonal and diagonal steps. If another piece is in its path, it may jump that piece like a standard chess knight, but must stop after the jump. This is the extended version of the knight-move.
The Lionheart King takes the place of the bent or extended Squire, and is the royal piece that must be checkmated to win. It takes its name from the lion from Chu Shogi and Dai Shogi**. It moves as a standard chess king, but as a half action, and a player may make one or two half actions in a single turn. It is thus able to capture two pieces in a turn, or capture and move back to its starting square (the latter called igui), though there are restrictions. It cannot end its turn on a threatened square (same as standard chess), it cannot capture on or move through a threatened square to capture unless it captures the only piece threatening that square ("check en passant"), it cannot take its second step if it captures a pawn on the first step ("pawn stalling") and if it starts its move in check it may only take one step unless the second step would capture the only piece giving check ("staring down"). There is no castling; the king is much more mobile in this variant, so there seems to be less need (also, the king and rooks start on different ranks).
Pawns have the same basic move as in standard chess: moving passively one step directly forward, and capturing one step on either forward diagonal. There are a few differences, however. When sitting on the third rank, it can also move passively one step diagonally forwards, but loses this ability as soon as it has left that rank. A pawn move is a half-action, like the Lionheart King's move, but a single pawn cannot be moved twice on a single turn (in other words, you can move two pawns per turn). The only exception is if the pawn has not moved on a previous turn: in that case, it may make two passive moves. A pawn that has made a double move as its first move may be captured by an enemy pawn on the opponent's following turn as if it had only made the first of its two moves (en passant capture). No more than one pawn may capture on a single turn.
To balance first move advantage, white's first pawn move is a full action.
Pawns may promote when they reach the 9th or 10th rank. A pawn may only promote to a piece that has been previously captured by the opponent. Promoting a pawn is a half action, with two restrictions: no more than one of each player's pawns may promote in a turn, and a promotion cannot be the only action taken (at least one half action must move a piece). All of the pieces that a pawn may promote to move as a full action, so it's not possible on a single turn to promote a sitting pawn and then take the promoted piece's move.
Half actions may be mixed. For example, in a single turn a player may move two pawns, or move and promote one pawn, or move a pawn and the king (one space), or move the king two spaces, or move the king one space and promote a pawn.
I haven't really decided on a board setup yet. I think I want to keep the king and queen next to each other in the middle. There are 4 2nd-rank piece types and 10 files, so there will be four of one piece on that rank (probably the guards; they're a bit cannon-foddery) and two of each of the others.
*In his article on bent riders on chessvariants.org, Ralph Betza names this piece the "Aanca", but I don't like that name. It's just Spanish for "gryphon". I prefer Hippogriff because it is like a gryphon, but also like a horse (knight): its move includes the Xiangqi horse (mao, a knight that cannot jump), and it always ends up on a square of the color opposite the one it started on. CV.org uses "hippogriff" for a limited form of the Gryphon, but they seem unlikely to coexist in a single game.
**Old, larger variants of shogi. Chu shogi ("middle shogi") was played with 46 pieces per side on a 12x12 board. Dai shogi ("large shogi") used 65 pieces per side on a 15x15 board. Sho shogi ("small shogi") was played on a 9x9 and evolved directly into modern shogi.