R_(1-tt)^(SN)(NP) Distinguishes Robust Many-One and Turing Completeness

Do complexity classes have many-one complete sets if and only if they have Turing-complete sets? We prove that there is a relativized world in which a relatively natural complexity class-namely a downward closure of NP, \rsnnp - has Turing-complete sets but has no many-one complete sets. In fact, we show that in the same relativized world this class has 2-truth-table complete sets but lacks 1-truth-table complete sets. As part of the groundwork for our result, we prove that \rsnnp has many equivalent forms having to do with ordered and parallel access to $\np$ and $\npinterconp$.