  The elastic Rayleigh scattering of twisted light and, in particular, the
polarization (transfer) of the scattered photons have been analyzed within the
framework of second-order perturbation theory and Dirac's relativistic
equation. Special attention was paid hereby to the scattering on three
different atomic targets: single atoms, a mesoscopic (small) target, and a
macroscopic (large) target, which are all centered with regard to the beam
axis. Detailed calculations of the polarization Stokes parameters were
performed for C^{5+} ions and for twisted Bessel beams. It is shown that the
polarization of scattered photons is sensitive to the size of an atomic target
and to the helicity, the opening angle, and the projection of the total angular
momentum of the incident Bessel beam. These computations indicate more that the
Stokes parameters of the (Rayleigh) scattered twisted light may significantly
differ from their behaviour for an incident plane-wave radiation.
