Yes i would say opposites are usually metapysical. The world is better described in sliding scales then opposites i think.

I mean not a whole lot of things are true opposites in the world, it's usually only language that wants to take us there...

For instance, truth and falsity, reason and emotion... are not true opposites.

So to finally answer your question :-), I guess purely logically or mathematically, the opposite of the opposite is sameness (or identity).

X = - (-X)

Edit : Although this is merely negation, not sure if this is the same, or if there is an equivalent for opposites in logic and math. What does one mean with opposites exactly anyway?

Opposition requires a context of agreement - up and down are similarly directions under gravity, and this sameness is what allows their opposition. Further,the opposed pair up/down is itself opposed to sideways. The opposite of a fish is not a bicycle, though man can be opposed to woman. And in another context, man can be opposed to nature, or to animal, or to mouse, or God.

To declare an opposite devoid of context of sameness is to fall in to a Venn diagram world in which any not-X is the opposite of X. And in this context, a fish is indeed the opposite of a bicycle, but so is common sense or a ripe camembert.

In classical logics yes, in paraconsistent logics no:

"Classical rules which govern the valid procedures for assigning the values of true and false is expressed in the Aristotelian idea that the negation of the negation of A (not not-A) yields A. However, for paralogicians, it is possible that the negation of the negation of A does not yield A, or rather that it could yield A and something else. Double negations have strange properties in paralogic, but we have some familiar examples of their effects in our ordinary linguistic usage. For example, one of the characteristic rhetorical tropes of Wordsworth's poems is litotes or understatement, which often makes use of the double negation, as in "I am not unwilling." We know what it means to be willing and unwilling, but we also know that to say "I am not unwilling" is nowhere near the same thing as saying "I am willing." The Greimasian semiotic square produces an excessive term at the place of double negation, and [the] example of the "undead" is another case of it". (Rothenberg, The Excessive Subject).

To declare an opposite devoid of context of sameness is to fall in to a Venn diagram world in which any not-X is the opposite of X. And in this context, a fish is indeed the opposite of a bicycle, but so is common sense or a ripe camembert. — unenlightened

So, we've been describing things as if they exist in Venn diagram substrates. Is this the point here? That we talk about things as if they were black and white, all the time. It's just such a subconscious process that nobody really takes notice?

So to finally answer your question :-), I guess purely logically or mathematically, the opposite of the opposite is sameness (or identity). — ChatteringMonkey

Yes; but, if all is one, then there's no identity apart from the whole. Sounds about right?

We know what it means to be willing and unwilling, but we also know that to say "I am not unwilling" is nowhere near the same thing as saying "I am willing. — StreetlightX

Why do they mean different things if they denote the same thing, then, if I may ask? Is this a prototypical example of something possessing a sense?

So to finally answer your question :-), I guess purely logically or mathematically, the opposite of the opposite is sameness (or identity). — ChatteringMonkey
Yes; but, if all is one, then there's no identity apart from the whole. Sounds about right? — Posty McPostface

Parmenides would agree.

Why do they mean different things if they denote the same thing, then, if I may ask? — Posty McPostface

'Different', 'same', 'mean', 'denote'; these are all fraught terms if you're not clear what is being understood by them. The thing is, there's no answering your OP in the abstract. In itself the question is more or less meaningless - an 'idling engine' of language kinda deal. It's only in reference to what one is trying to do with the language one uses that the question can 'take on' sense. What motivates it? What context it is being employed in? Sans context sans sense.

"For instance, truth and falsity, reason and emotion... are not true opposites." How are truth and falsity not opposites?