The inviscid instability of heterogeneous swirling flows with radius-dependent density is investigated and secular relations for the instability growth rates for several different flow configurations are obtained from explicit solutions of the governing equations. It is found, in agreement with a sufficiency condition for the stability of such flows obtained earlier, that they are stable to both axisymmetric and non-axisymmetric infinitesimal modes whenever the density is a monotonic increasing function of radius and at the same time the radial variations in both the angular and axial velocity components remain small. The instability mechanisms present in these flows are both of centrifugal and of shear origin, the classical Rayleigh–Synge criterion being a condition for centrifugal stability. It is shown, via several counter examples, that the Rayleigh–Synge criterion for the stability of swirling flows is generally neither a necessary nor a sufficient condition when non-axisymmetric disturbances are considered or large shears exist in the flow. Very stable flows occur when the angular and axial velocity components have no radial variation and simultaneously the density increases with radius as is the case in a typical centrifuge.