I just wanted to confirm this with someone.
When we refer to reflections of the chessboard about the diagonal, we are not talking about rotations (which would assume that the chessboard has the same face on the bottom and top). Instead, if use a matrix to represent the chessboard, we are referring to swapping the tiles (entry $(i,j)$) on the chessboard with their counterparts on the other side of the diagonal (with $(j,i)$).
Is this true?
A rotation is never a (single) reflection; the former is orientation preserving, and the latter isn't. And yes: if you use a matrix to represent the board, a reflection across the northwest-southeast diagonal corresponds to a transposition of the matrix.