Complexity of Turing self-reducibility of Clique problem?
This is not a reference, but an NP-complete problem cannot have a polynomial-time self-reduction which halves the input size unless NP has a nO(log n)-time algorithm, even if we allow Turing reductions.

Suppose there is a p(n)-time Turing reduction from an NP-complete problem L to itself such that given an input of length n, the reduction only makes queries to strings of length at most n/2.  Then you can convert this reduction to a standalone deterministic algorithm for L by using recursion.  By writing down a recurrence relation for the running time of this deterministic algorithm, it is not hard to see that it runs in time nO(log n).