  Vortex electrons, - freely propagating electrons whose wavefunction has
helical wavefronts, - could become a novel tool in the physics of
electromagnetic radiation. They carry a non-zero intrinsic orbital angular
momentum (OAM) $\ell$ with respect to the propagation axis and, for \ell \gg 1,
a large OAM-induced magnetic moment, \mu ~ \ell \mu_B (\mu_B is the Bohr
magneton), which influences the radiation of electromagnetic waves. Here, we
consider in detail the OAM-induced effects by such electrons in two forms of
polarization radiation, namely in Cherenkov radiation and transition radiation.
Thanks to the large \ell, we can neglect quantum or spin-induced effects, which
are of the order of \hbar \omega/E_e \ll 1, but retain the magnetic moment
contribution \ell \hbar \omega/E_e \lesssim 1, which makes the quasiclassical
approach to polarization radiation applicable. We discuss the magnetic moment
contribution to polarization radiation, which has never been experimentally
observed, and study how its visibility depends on the kinematical parameters
and the medium permittivity. In particular, it is shown that this contribution
can, in principle, be detected in azimuthally non-symmetrical problems, for
example when vortex electrons obliquely cross a metallic screen (transition
radiation) or move nearby it (diffraction radiation). We predict a left-right
angular asymmetry of the transition radiation (in the plane where the charge
radiation distributions would stay symmetric), which appears due to an
effective interference between the charge radiation field and the magnetic
moment one. Numerical values of this asymmetry for vortex electrons with E_e =
300keV and \ell = O(100-1000) are O(0.1-1%), and we argue that this effect
could be detected with existing technology. The finite conductivity of the
target and frequency dispersion play the crucial roles in these predictions.
