  We calculate the twisted Hochschild and cyclic homology (in the sense of
Kustermans, Murphy and Tuset) of the coordinate algebra of the quantum SL(2)
group relative to twisting automorphisms acting by rescaling the standard
generators a,b,c,d. We discover a family of automorphisms for which the
"twisted" Hochschild dimension coincides with the classical dimension of SL(2,
C), thus avoiding the "dimension drop" in Hochschild homology seen for many
quantum deformations. Strikingly, the simplest such automorphism is the
canonical modular automorphism arising from the Haar functional. In addition,
we identify the twisted cyclic cohomology classes corresponding to the three
covariant differential calculi over quantum SU(2) discovered by Woronowicz.
