Shouldn't it be even because to every position there is a mirrored position, mirrored on the axis between the e and d files. (Note that a position cannot be mirror symmetric because of the kings.

@user1583209 the mirror image of the initial position isn't exactly legal (or at least the OP should clarify whether it is) - it cannot be reached from the starting position.

@Glorfindel: Good point. Is this (and similar positions where queen/king could not have moved) the only exception?

Clearly there has to be some matching strategy based on symmetry, but flipping left/right isn’t viable I think, because it’s hard to marshal the exceptional positions.

@bot I think the approach would be to count the positions that cannot be reached legally with the opposite color on the move. For example, you can reach the equivalent of 1. e3 with 1. Nf3 e6 2. Ng1 Bd6 3. Nf3 Be7 4. Ng1 Bf8, so there are an even number of legal positions from that point. Of white's 20 opening options, only the four knight moves, a3, f3, and h3 have unreachable equivalents with black. I think it could still be very difficult, though.

@DKrueger Unfortunately, that includes not just opening positions, but every position which includes a check.

@bof it’s a good sub-puzzle so I won’t give the answer - it’s only a corner case. Dealing with castling properly is a necessary first step.

@bof This is a question I want the answer to because I don’t know it. But I have some partial results and I want you guys to reach the frontier of my thinking. One cute thing I have found is this e.p. idea: someone else might enjoy looking for it. You need to think about the different sets of castling rights which are possible and the implications so that you don’t double count. I am only doing matching based on non-vampire mirroring. I don’t see any other way to be sure that the matching is 1-1. Hope this helps

@bof attitude? What attitude? I’m baffled

@Laska There is some strange interaction between castling and parity which I can't quite see how to count effectively yet. Take for instance the position after 1. Nf3 Nf6 2. Ne5 Ng8 3. Ng6 Nf6 4. Nxf8 Ng8. This position cannot be "mirrored" (inversion about horizontal axis + colour inversion) because of the side to move, despite a piece being captured -- you can't reach the "mirror" position while preserving white kingside castling rights.

Uncooperative. Anyway, thanks for posting that position. Very pretty! But unimportant to the even/odd question. Whew!

@Remellion: I think there are 16 independent vampire clans, each for a different combination of castling rights. The head of each clan has the pieces all in game array squares. If a player has one castling right then the only way they can move K and R is by castling. Any other move would duplicate a position in one of the other less entitled clans

@bof so is the number of en passant vampires odd or even? :)

Even, of course..

@bof Yes I think so too. Two possible double pawn moves in every case. If as was suggested we count all the vampires, then we should count all of these little guys separately

If I have to rely on humanly identifying matrices like the one I posted to do a census of the clans, then I wouldn't call that "effective". :P Too easy to miss an idea or wrongly identify a corner case.