The answer is no under the following assumptions. Some of them do only apply to very strong players. If you define things a bit different you might get different results. So this is of course not the last and definite answer to the question!
1.: An optimal strategy in chess aims to improve the own situation which means improve some score S(t) (which I will leave to you) or prevent the other side from improving the situation (aka the score) in the future. Improving the situation means, that your new situation keeps the result of the actual situation in case both parties follow the best strategy (here are the strong players) while making it more likely to win (or not lose in case you currently have the lower score) if the other player does not follow the optimal strategy.
2.: The score does only depend on the figures on the board and not on the psychological likelihood of the other side to give up, or to run over time etc.
3.: No pawns left.
4.: Your score is designed in a way, that given optimal play the party with the higher score would win the game unless both parties have the same score, which results in draw.
5.: A "realy good" chess player knows if a game is diverted and will share the game with each other.
Let S(t) be the score at the position, which should be mirrored. Lets assume the optimal strategy for both sides mirrors the situation at time S(t+m). Because our score does only take into account the position of the figures (see assumption 2) and all figures have the same possible moves from the mirrored position (see assumption 3) S(t)=S(t+m). The side which is aimed to win the game could have done so from the position at time t (because all figures have the same possible moves from the mirrored position). So we need a situation, where our optimal strategy for both sides from this situation is only preventing the other side from increasing the own score. This results in both sides to never increase the own score. So you need a situation which results in infinite repetition of the mirrored positions. After assumption 5 this will result in sharing the game being the best strategy. Which means that you will never see the mirrored situation.
I hope this helps you to define things in a way, that makes discussion about possible solutions with "yes" possible!
Disclaimer: Not a native speaker. I therefore welcome comments on language improvements or edits regarding this.