We study the autoreducibility and mitoticity of complete sets for NP and other complexity classes, where the main focus is on logspace reducibilities. In particular, we obtain:- For NP and all other classes of the PH: each logspace many-one-complete set is logspace Turing-autoreducible.- For P, the delta-levels of the PH, NEXP: each logspace many-one-complete set is a disjoint union of two logspace 2-tt-complete sets.- For PSPACE: each polynomial-time dtt-complete set is a disjoint union of two polynomial-time dtt-complete sets.