Let F denote the Fourier transform on L2(R), and let T ≡ DφMu, where Dφ ≡ FMφF−1, φ ∈ H∞(R) is inner, and |u| = 1 a.e. This paper gives a partial description of the spectral multiplicity theory of T . It is shown that T is absolutely continuous and is a bilateral shift of infinite multiplicity if φ is not a finite Blaschke product. Similar results are obtained for the (isometric) restrictions of T to the invariant subspaces L2(α,∞). Specifically, these restrictions always have absolutely continuous unitary parts, and shift parts with multiplicity equal to the multiplicity of φ.