Let $X$ be an algorithmic task. (It can be a decision problem or an optimization problem or any other task.) Let us call $X$ "on the polynomial side" if assuming that $X$ is NP-hard is known to imply that the polynomial hieararchy collapses. Let us call $X$ "on the NP-side" if assuming that $X$ admits a polynomial algorithm is known to imply that the polynomial hierarchy collapses.
Of course, every problem in P is on the polynomial side and every problem which is NP-hard is in the NP-side. Also, for example, factoring (or anything in NP intersection coNP) is on the polynomial side. Graph isomorphism is on the polynomial side. QUANTUM-SAMPLING is in the NP-side.
1) I am interested in more examples (as natural as possible) of algoritmic tasks in the polynomial side and (especially) in more examples in the NP side.
2) Naively it looks that the NP side is a sort of a "neighborhood" of the NP-hard problems, and the P-side is a "neighborhood of P". Is it a correct insight to regard problems in the NP side as "considerably harder" compared to problems in the P side. Or even to regard problems in the NP side as "morally NP-hard?"
3) (This might be obvious but I don't see it) Is there an $X$ on both sides or are there theoretical reasons to believe that such an $X$ is unlikely. Update The answer is YES; see Yuval Filmus' answer below.
(If these "sides" are related to actual complexity classes and if I miss some relevant cc jargon or relevant results please let me know.)
Update: There are by now several very good answers to the question. As noted first by Yuval Filmus and mentioned again the question is not formal and some restriction on the argument showing that X is on the P-side/NP-side is needed. (Otherwise, you can have X to be the task of presenting a proof for 0=1 which is on both sides.) Putting this aside, it may be the case that problems X (genuinly) on the NP-side capture somehow the hardness of SAT, although this may also be the case for some problems on the P-side where the hardness of SAT is weakened (even slightly) in a provable way. Yuval Filmus gave a weakened version of SAT which is on both sides. Andy Drucker gave (in two answers) five interesting examples including a reference to Schöning's Low and High hierarchies, and Scott Aaronson gave further interesting examples, mentioned the question of inverting a one-way function which is close to capture NP hardness and yet on the P-side, and his answer also discusses the interesting case of QUANTUMSAMPLING. I metioned an old result of this kind by Feige and Lund.