NP-complete: Language is NP-complete, when it is in NP and every problem in NP is reducible to it in polynomial time. But what languages are reducible to a NP-complete problem (for example SAT) in polynomial time - other than languages in NP.
NP-hard: A problem H is NP-hard when every problem L in NP can be reduced in polynomial time to H. This is reversed reducing, so I'm not sure if this is a good candidate. We don't know whether one can solve NP-hard problem in polynomial time.
If P ≠ NP, then NP-hard problems cannot be solved in polynomial time. (NP-hardness wiki)
So, my question is what class of languages can be transformed to a NP-complete problem in polynomial time using Turing reductions and/or many-to-one reductions.