  We calculate the one-photon loop radiative corrections to charged pion
Compton scattering, $\pi^- \gamma \to \pi^- \gamma $. Ultraviolet and infrared
divergencies are both treated in dimensional regularization. Analytical
expressions for the ${\cal O}(\alpha)$ corrections to the invariant Compton
scattering amplitudes, $A(s,u)$ and $B(s,u)$, are presented for 11 classes of
contributing one-loop diagrams. Infrared finiteness of the virtual radiative
corrections is achieved (in the standard way) by including soft photon
radiation below an energy threshold $\lambda$, and its relation to the
experimental detection threshold is discussed. We find that the radiative
corrections are maximal in backward directions, reaching e.g. -2.4% for a
center-of-mass energy of $\sqrt{s}=4m_\pi$ and $\lambda=5 $MeV. Furthermore, we
extend our calculation of the radiative corrections by including the leading
pion structure effect (at low energies) in form of its electric and magnetic
polarizability difference, $\alpha_\pi - \beta_\pi \simeq 6\cdot 10^{-4}
$fm$^3$. We find that this structure effect does not change the relative size
and angular dependence of the radiative corrections to pion Compton scattering.
Our results are particularly relevant for analyzing the COMPASS experiment at
CERN which aims at measuring the pion electric and magnetic polarizabilities
with high statistics using the Primakoff effect.
