A bishop's or queen's diagonal move reflects off all four edges of the board like a billiard ball. It stops at the first piece it meets (friend or foe), capturing if it's an enemy. A move that would return to its own starting square is not allowed.
Rooks, knights, kings, and pawns move exactly as in ordinary chess. The only change is the diagonal movement of bishops and queens: when the path hits an edge of the board, it reflects 90° and keeps going in a straight line — hitting a corner can reflect off both edges at once. The path stops the moment it reaches any piece: blocked by a friendly piece, or capturing and stopping on an enemy piece.
Simplifications: castling and en passant are omitted. Check and checkmate aren't enforced — capturing the king directly ends the game.