We reanalyze the problem of radiative polarization of electrons brought into
collision with a circularly polarized strong plane wave. We present an
independent analytical verification of formulae for the cross section given by
D.\,Yu. Ivanov et al [Eur.\ Phys.\ J. C \textbf{36}, 127 (2004)]. By choosing
the exact electron's helicity as the spin quantum number we show that the
self-polarization effect exists only for the moderately relativistic electrons
with energy $\gamma = E/mc^2 \lesssim 10$ and only for a non-head-on collision
geometry. In these conditions polarization degree may achieve the values up to
65%, but the effective polarization time is found to be larger than 1\,s even
for a high power optical or infrared laser with intensity parameter $\xi =
|{\bf E}| m c^2/E_c \hbar \omega \sim 0.1$ ($E_c = m^2 c^3/e \hbar$). This
makes such a polarization practically unrealizable. We also compare these
results with the ones of some papers where the high degree of polarization was
predicted for ultrarelativistic case. We argue that this apparent contradiction
arises due to the different choice of the spin quantum numbers. In particular,
the quantum numbers which provide the high polarization degree represent
neither helicity nor transverse polarization, that makes the use of them
inconvenient in practice.Comment: minor changes compared to v3; to appear in PR