Let us assume we live in a shared Universe which can be fully described mathematically, and that all mathematical variables have an opposite. 2 is opposite to -2, and true is opposite to false. When I was a child we played "opposite day", where we interacted as usual but everything had to be expressed and interpreted as its opposite. If everything is opposite, how will the world look? Will everything ultimately be the same, because the function of event handlers are reversed along with it's input?
In light of Robert's comment, I'm extending this answer in one respect. The thought experiment can be understood in two ways.
Way 1: Linguistic opposites: We are merely changing the meaning of words. "2" means -2. "-2" means 2. The utterance "true" now means false and vice versa. This particular thought experiment is completely uninteresting, because it merely means that the language of this opposite world is somehow coincidentally reversed for all descriptive terms.
Way 2: Metaphysical opposites: Here, the claim is that what is true in our world is false in this world, and what is false in this world is true in that world. In other words, it's not that they say "2" when they mean -2, it's that they use language in the same way but the objects that populate their world are opposite.
But this is a self-defeating thought experiment, because when we move past trivial elements (like positional functions), not everything can admit of opposites in the way you're describing and that's what basically kills the thought experiment. Sure 2 and -2 can be opposites, but true and false differ not just as poles but as functions that relate to reality. E.g., I am bunny and I am a tarantula are both false. If you reverse the meanings of true and false, then they both become true which is self-contradictory.
What this does show us, however, is that the units for position functions are arbitrary. (We can put the origin (0,0,0,...) wherever we want and just move everything from there). Mass functions and many other types of evaluative functions are not. To give an example:
Is is true that I am wearing a shirt and it is white? Is is true that I am wearing a shirt? Is is true that I am wearing a white shirt?
Consider if I am wearing a blue shirt under normal and opposite evaluation. Under normal, evaluation: false, true, false. Under opposite evaluation? it's not at all clear. Do statements containing and become true if either term is true in opposite evaluation? Do we apply the opposite afterward fully evaluating? (i.e. if we fully evaluate and then opposite: we get true, false, true. If we evaluate each piece and then opposite without changing the logical operators: false, false, true)
The CPT Symmetry theory in quantum mechanics might answer your question. It states that if you flip the charge of every particle (C), invert the parity (P) of the universe (reflect the physical coordinates of space (x,y,z) becomes (-x,-y,-z)), and reverse time (T), then the new universe would be indistinguishable from the current universe.
Logically and mathematically, there is no difference; what you are considering is called an isomorphism; Physically of course there is plenty of difference.
Were you to take the 'universe' of Set, the class/category of all sets, and consider the universe of -Set where all morphisms, that is functions between sets are reversed. Then one can show that Set & -Set are not identical but are equivalent.
When however your consider, our physical universe Omega and reverse physical notions in it, -Omega;its obvious to see that they are very different.