Do many-one reductions and Turing reductions define the same class NPC I wonder if NPC classes defined by many-one reductions and Turing reductions are equal.

Edit:
Another question, are Turing reductions only collapsing C and co-C classes for some C or is there a class $C$ such as there exists a problem not in $C \cup co-C$ under Karp reduction and which is in $C$ under Turing reduction ?
The "Boolean Hierarchy" BP is a whole hierarchy of combinations of NP problems under Karp reductions which are all in P^NP.