I wouldn't discount symmetries in chess on the account of algebraic topology existing.

Once in a while, new maths is produced in one area, and somebody else would pick it up to apply in a completely different domain. I'm a bit familiar with chess engines, & their source code. The board representation is just cache-friendly data structures of a 8x8 board, and heuristics on top. This is only a statement on our understanding: heuristic, or probabilistic (see AlphaZero), but it doesn't mean a more fundamental, symmetrical structure doesn't exist. Rubik's cube was famously solved by fundamental insights from group theory. Now, chess is probably, but not definitely, radically harder problem in terms of computational complexity, let alone because there's an adversary and all of game logic applies. However, we see it all the time in cryptanalysis where new insights from maths people broke some very cool constructions out-of not much but some bad priors.

Pure min-max search is inherently suboptimal, if your goal is understanding the game. AlphaZero, and to a lesser extent Leela et al. has shown this, and indeed the players incorporated all these ideas shortly thereafter. Of course, old tricks no longer provide advantage now that they're known, but then again—it doesn't mean better interpretations in commentary, player training, etc. are not possible. None of the existing engines, heuristic, probabilistic, heuristic and probabilistic, are so far (a) bringing new maths to help apply new maths to chess positions, (b) bringing new game-representations that would lend better to interpretability during inference.

To truly solve chess, in a given engine both (a) and (b) must suffice.

To get +EV from some intricate line analysed as deep as reasonable in the allotted time is not to bring you any closer in understanding the game. You could tell that the line works, of course, but that would only be possible on account of you already having found the line in the first place! However, for the line to be possible, and profitable, something in the structure of your prior position has to allow it. Or otherwise predict it.