  We present a theoretical investigation of the radiative recombination of
twisted Bessel electrons with initially hydrogen-like (finally helium-like)
heavy ions. In our study, we focus especially on the linear polarization of
x-ray photons emitted in the electron capture into the ground $1s^2_{1/2}$
ionic state. Particular emphasis is placed on the question of how this
polarization is affected if incident hydrogen-like ions are themselves
spin-polarized. To explore such a "polarization transfer" we apply the density
matrix theory and derive the Stokes parameters of recombination x-rays for the
realistic case of collisions between macroscopic electron and ion beams. Based
on the developed general approach two scenarios are discussed that are of
interest for the planned experiments at ion storage rings. First, we
demonstrate how the use of twisted electrons can empower the known method for
the diagnostics of spin-polarized ion beams, based on the rotation of the
linear polarization of recombination light. In the second scenario we show how
the internal structure of ions beams with inhomogeneous intensity and spin
patterns can be probed by the capture of Bessel electrons, carrying different
values of angular momentum.
