Twisted Gaussian Schell-model beams were introduced 25 years ago as a celebrated example of a "genuinely two-dimensional" partially coherent wavefield. Today, a definite answer about the effect that a twist phase should produce on an arbitrary cross-spectral density has not yet been reached. In the present Letter, the necessary and sufficient condition for a typical Schell-model partially coherent CSD endowed with axial symmetry to be successfully mapped onto a bonafide twisted CSD is addressed. In particular, it is proved that any shift-invariant degree of coherence of the form μ(|r-r|) is "twistable" if and only if the zeroth-order Hankel transform of the radial function μ(r)exp(ur/2) (with u being the twist strength) turns out to be a well-defined, non-negative function.
