The cozy neighborhoods of "P" and of "NP-hard" 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.
The very terms "on the P-side" and "on the NP-side," and of course the question title, encourage us to imagine a "cozy neighborhood" surrounding P and another "cozy neighborhood" surrounding the NP-hard problems.  However, I'd like to argue that these two neighborhoods are not so "cozy" at all!

As a first observation, there are problems "on the P-side" that seem "morally" much closer to NP-hard than to P.  One example, anticipated by Gil of course, is the general problem of inverting one-way functions (depending on exactly what type of reductions are allowed; see Bogdanov-Trevisan or Akavia et al.).

Conversely, there are also problems "on the NP-side" that seem "arbitrarily far" from being NP-hard.  One silly example is a random language L, with probability 1 over L!  For if such an L is in P, then 0=1 and math is inconsistent, and therefore PH collapses also. ;-D

(Note that a random language L is also "on the P-side," with probability 1 over L.  For almost all such L's have the property that if they're NP-hard, then NP&sube;BPP and PH collapses.  And this gives a proof, much simpler than the appeal to Ladner's Theorem, that there exist languages on both "sides."  Indeed, it shows that of the uncountable infinity of languages, "almost all" of them -- in fact, 100% -- are on both sides!)

This sounds like juvenile game-playing, but there's a serious lesson I'd like to draw from it.  I'd argue that, even though QUANTUM SAMPLING is formally "on the NP-side," that problem is barely any closer to being "morally NP-hard" than the random language L was.  Arkhipov and I (and independently, Bremner-Jozsa-Shepherd) showed that, if QUANTUM SAMPLING is in P (or rather, in SampBPP, the class of polynomially-solvable sampling problems), then P#P = BPPNP, and therefore the polynomial hierarchy collapses.  Yet if you're a BPP machine, an oracle for BosonSampling would, as far as we know, bring you no closer to solving NP-complete problems than a random oracle would.  Only if you already have the ability to solve NP-complete problems -- say, if you're a BPPNP machine -- do you "notice" that the BosonSampling oracle boosts your capabilities even further, to #P.  But the property of boosting NP up to #P seems different than, and maybe even "orthogonal to," the property of being NP-hard on one's own.

Incidentally, a wonderful open problem suggested by Gil's question is whether BosonSampling is also "on the P-side."  That is, can we show that if NP reduces to BosonSampling then PH collapses?  While I might be missing something obvious, at first glance I have no clue how to prove such a thing, any more than I know how to prove the stronger implication that if NP &sube; BQP then PH collapses.