This problem comes from a study of game design, specifically the wonderful abstract strategy game Kamisado, which I highly recommend for the mathematically minded.
The rules of the game are not so important, I am interested only in the board, which looks like this -
The board is an 8x8 Latin Square. I wonder how much choice the designer had in choosing the board. The only restrictions we want are that the board should be the same for both players, and that swapping two colours doesn't matter in terms of gameplay.
So the question is, more generally, "How many Latin squares of size $n$ exist which are isomorphic under a rotation of 180 degrees and symbol permutation?"
Specifically, for $n=8$, "How many possible Kamisado boards are there?". I know the answer must be in the literature somewhere, but am unfamiliar with the area and terminology. If there is a similar question somewhere please point me towards that instead.