  We present a theoretical investigation of the resonant elastic scattering of
laser photons by ultra-relativistic partially stripped ions, that is the core
process of the Gamma Factory project. Special emphasis in our study is placed
on the angular distribution and polarization of scattered photons as observed
in the collider and ion-rest reference frames. In order to describe these
(angular and polarization) properties for arbitrary relativistic many-electron
ion, the general approach, based on the application of irreducible polarization
tensors, is laid down. By making use of the polarization tensors we explore in
detail the scattering of both, conventional plane-wave- and twisted (or vortex)
photons. For the former case we show how the propagation directions and
polarization states of incident and outgoing photons are related to each other
for the $n S_{0} \to n' P_{1} \to n S_{0}$, $n S_{1/2} \to n' P_{1/2} \to n
S_{1/2}$ and $n S_{1/2} \to n' P_{3/2} \to n S_{1/2}$ resonant transitions. For
the scattering of initially twisted light, that carries non-zero orbital
angular momentum, we explore the angular distribution of secondary photons and
discuss the conditions under which they are also twisted.
