Chess480: Why 480?

Question

From wiki:

Lewis has named this chess variation "Chess480"; it follows the rules of Chess960 with the exception of the castling rules. Although a Chess480 game can start with any of 960 starting positions, the castling rules are symmetrical (whereas the Chess960 castling rules are not), so that mirror-image positions have identical strategies; thus there are only 480 effectively different positions. The number of starting positions could be reduced to 480 without losing any possibilities, for example by requiring the white king to start on a light (or dark) square.

Ok someone tell me the maths here please. Why do the castling rules make this 480 from 960?

Answer 1

Consider two starting position that are each other's horizontal mirror images. Say, the normal chess starting position, and the one with king and queen reversed.

In chess 480, those two are essentially the same game, normal chess: in normal chess you can castle long with the king ending on c1, and short with the king ending on g1. In the reversed position you can castle long with the king ending on f1 and short with the king ending on b1; as a result, any position resulting from the chess starting position is equivalent to a position from the reversed chess position, every possible move and strategy has a direct reversed equivalent. They're the same.

In chess 960, in the reversed starting position, long castling still ends up with the king on c1 and short with the king on g1, and that is different from normal chess position because the queen started on e1 instead of d1.

So chess 480 has half the positions of 960 because they are pairwise equivalent due to symmetry.

Answer 2

In Chess960, after castling, the kings and rooks end up in the same place as they do in standard chess. That is, if white castles kingside, the king will always be on g1 and the rook will always be on f1.

In Chess480, my understanding from your quote is that if the white king starts on e1, f1, or g1, then after castling kingside, it will end on g1 and the rook will be on f1. However, if the king instead starts on b1, c1, or d1, then castling kingside is castling "long", and the king will end up on the f1 square, and the rook will end on the e1 square.