doubtful, or at least not useful ones. Like, you could describe some invariant along the lines of "the position is winning for the side-to-move, iff there exists move, such that position' := ApplyMove(position, move) is losing for the (now other) side-to-move". But that's just restating minimax algorithm that people have known for 50 years.
As someone dabbling abit around chess engine development, I'm very often impressed by the many intricacies and observations made by people who pushed the envelope. It just doesn't sound plausible people wouldn't have discovered these killer invariants by now if they existed
I agree it's hard and non-obvious. If it wasn't then chess would have long been solved by now.
Let's start from the other end. Just a pawn and two kings. It's possible to describe some quite succinct rules for when that's a draw versus a win for the side with the pawn. Agreed? Club players know these by heart. You could write that doen as invariants. As long as the side with the pawn stays inside the "green zone" of the state space, there is nothing the other side can do to void mate. And vice versa, if the game is in the red zone and the other player manages to stay inside that red zone, there is nothing the side with the pawn can do to win. Those areas of the state space, green and red zones, can be described as invariants, in contrast to just enumerating them. It's very compact and can easily be checked by a machine that it's correct.
Now let's add a pawn. And another. And a rook perhaps. The more you add, the harder the condition is to describe, but we live in the age of billion-node-sized neural nets, we have the resources. Eventually you get all pieces on the board, and if the starting position satisfies the draw invariant, that's it. And likely the 960 freestyle chess positions too.
You may want to take a look at the Shannon number, we would need quite a large neural net to solve chess in this way.