I'm not sure where to look for an analysis of this. Perhaps I can just use an "either/or" state of affairs in a conversational context?
You may say you're sorry or you may not. I couldn't
care less either way.
Here, the alternatives indicate the two possible courses of action (two narratives, if you like) which lead to "...either way". The first option (saying sorry) shows contrition. The second (not saying sorry) is intransigent. So not saying sorry in this case is as conscious an action as saying it.
To say "You may say you're sorry. I couldn't care less" is to miss out the "action" of not saying sorry and the speaker's response to it. (And the statement can hardly continue with "either way"). So there's no redundancy here.
Further, to say "You may say you're sorry and you may not say you're sorry" is to insist on a paradoxical state for the person being talked to. The two actions are incompatible.
This is an artificial situation I've posed, and I can't think through all possibilities of using "may or may not", but I think in everyday usage the phrase can be used in consideration of two possible contrasting but equally valid states, with the result being the same in either case.
The question may or may not be, but probably is, a duplicate of this one, with more answers and points of view.