I invented this game rather to show history of chess than making something
playable:
Files A, B, C and D are 'west', and E, F, J and H are 'east'. In
beggining both players have pieces from chaturanga: chariot instead rook
(moves 1 or 2 squares orthogonally), elephant instead bishop (leaps exactly
2 squares diagonally), ferz instead queen (1 square diagonally). King and
knight are same. Pawns also same, but can't make duble step. When pieces
crossing center of board (between 4th and 5th ranks), the promotes: If they
entered 'west', they promotes to thier relatieves of modern european
chess (elephant promotes to bishops, chariots to rooks, ferz to queen,
other pieces don't change farther); if they entered 'east' after
crossing center, they promotes to thier shogi relatives (elephant to silver
generals, chariots to lances, ferz to gold general, pawns to shogi pawns
(captures also orthogonally forward), and knights loses thier power: now
they can move only forward, like shogi knight; king don't promote). All
shogi pieces (expect for king an gold general) can promote to gold general.
Promotion zone is on 2 last ranks (actually, in first variant of shogi 8X8
promotion zone already was 3 last ranks, but here 2 will be better).
Promotion rules are same as in shogi. Western pawns promotes on last rank,
only to queen, rook, bishop or knight.
Goal is to checkmate king. What will be after stalemate, it's chosen by
players before game.
As it is clear what western pieces are much stronger than shogi pieces, you
can can chose one or two of that advantages of shogi pieces:

1. When pieces are captured by shogi, They can be dropped (like in shogi).
There two variants (chose one before playing): first - captured chaturanga
and chess pieces becomes thier relatives from shogi (elephants and bishops
becomes silver general, etc.); second - they don't become shogi (if pawns
wich crossed center on west dropped to 2nd rank, they can make duble step
and can be captured en passant). Promoted western pawns becomes pawns again
in both variants. Dropping rules are same as in shogi.
2. It can be combined with 1. When at least four pieces are promoted in
east, player will get in reserve shogi pieces wich are called 'bishop'
and 'rook' by chess players. They both can be dropped on player's 2nd
rank in one turn. This turn must'nt check, checkmate or stalemate enemy
king. They can be promoted to dragon king and dragon horse as other shogi
pieces in this game.
3. It can be combined with 1, but can't with 2. Pieces don't become chess
or shogi forever. When they are in 'west' (after crossing center), they
are chess pieces, and when in 'east', they are shogi pieces. They can't
become chaturanga again. Promoted pieces becomes relatives of thier
PROMOTED forms (queens always becomes gold generals, etc.).

Did you like this game? Have you questions?
I also have variant of this game, where are chinese and mongolian zone.
I'll write about it later.