We present a number of results relating to the difference of two Gaussian Schell-model cross-spectral densities (CSDs). They allow us to specify conditions under which such a difference represents itself in a valid CSD. In particular, a sufficient condition is derived for the non-negative definiteness of the resulting CSD, for any admissible choice of the involved parameters, while a necessary and sufficient condition is obtained for the case of CSDs endowed with the property of being shape-invariant upon propagation.