For an irrational rotation α of the circle group T=R/Z and a piecewise absolutely continuous function f@[ratio ]T→R, the unitary operator Vh(x)=e2πif(x)h(x+α) on L2(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is κ-weakly mixing for some κ with 0<[mid ]κ[mid ]<1. In particular V has continuous singular spectrum. The property of κ-weak mixing (with possible change of the value of κ, 0<[mid ]κ[mid ]<1) holds for all irrational rotations and, given α, is stable under perturbations of f by functions with sufficiently small O(1/n)-norm. On the other hand, there exists a piecewise linear function f with two non-integer jumps such that the spectrum of V is continuous singular for one value of α and Lebesgue for another.