When is a non-self-adjoint Hill operator a spectral operator of scalar type?

We derive necessary and sufficient conditions for a one-dimensional periodic Schr\"odinger (i.e., Hill) operator H=-d^2/dx^2+V in L^2(R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V.