@MartinEnder What I'm imagining is a board where pieces are initially in some fixed positions and the only move you have is to rotate a row or column of the board, Labyrinth-style (possibly shifting a new tile into the board, instead of a proper rotation). The board could either really be moved underneath the pieces or the pieces could be moved along with the board.
If the board moves underneath the pieces, there'd have to be some meaning to the tiles. I could imagine either colours, so that pieces on the same colour can interact, or movement instructions which indirectly move the pieces.