I played around with this idea. First transforming the visual representation of the chess board and then seeing how I would describe the moves of the pieces and whether it was more / less intuitive.

It is easy to view the board as 64 consecutive spaces (left-to-right through the ranks, then snaking through the ranks, and then a circular representation. The circle was cool because the moves became rotations (a rook can stay in its octant or move an exact multiple of 45 degrees...). I did some 2D transformations with a one space skew on each rank (this made the bishop act simultaneously like a rook and knight which was interesting).

So it is easy to preserve the rules of chess and create a different visual representation of the board. Of course, every one I tried just made the game more difficult to understand. I wonder if there is a transform that could aid in understanding chess.