  The notion of non-equatorial spherical photon orbits is among the very
special properties of the Kerr spacetime of rotating black holes and is one
that leaves a clear mark on the electromagnetic and gravitational wave
signature of these objects. In principle, one could use observations like the
shadow a black hole casts when is externally illuminated or the gravitational
quasi-normal mode ringdown of merging black holes in order to identify the Kerr
metric via its light-ring and spherical photon orbit structure, or perhaps look
for the presence of more exotic non-Kerr ultracompact objects. This approach
would require some understanding of how circular photon orbits behave in
alternative (and far less special) non-Kerr spacetimes. In this letter we
explore the connection between the existence of spherical photon orbits and the
separability of a general stationary and axisymmetric spacetime. We show that a
spacetime cannot be separable if it possesses an equatorial photon ring but at
the same time does not admit (in any coordinate system) non-equatorial
spherical photon orbits. As a result, the separability-circularity connection
could serve as a non-Kerr diagnostic of black hole candidates.
