Well, certainly, (assuming continuous variation of such things), the IVT would let us get a pair of antipodal points with exactly the same temperature, and similarly a pair of antipodal points with exactly the same pressure. In fact, we could get such “opposite points with same values” pairs along any loop we like, for any particular single-dimensional quantity we like. But to get both temperature and pressure to be the same simultaneously? This would seem, to me, to require tools more sophisticated than the IVT (the Borsok-Ulam theorem, for example).

[The example is of some nostalgic significance to me because I have a very clear memory from high school of being told just this statement, trying to prove it from the IVT (after a sudden breakthrough in biology class on the simple way to do so for the 1-dimensional version), struggling, and finally thinking I could munge it together with the aid of certain “intuitive, but somewhat dodgy” leaps of logic, forcing me to ponder “Hm… Are these leaps legitimate or not? They seem rightish… but then, there’s no point in having something merely proofish, is there?”. Indeed, the leaps were ultimately untrue. I still don’t know if there’s a nice way to get it from just the IVT, though I’d be interested to see one.]