  The polarization of recoil photon ($\gamma'$) in the non-linear Compton
process $e + \vec L \to \vec \gamma' +e'$ in the interaction of a relativistic
electron with a linearly polarized laser beam ($\vec L$) is studied within the
Furry picture in the lowest-order, tree-level S matrix element. In particular,
we consider the asymmetry of differential cross sections ${\cal A}$ for two
independent axes describing the Compton process equal to the intrinsic spin
variable ${\xi}^f_3$, that determines the polarization properties of $\gamma'$.
The sign and absolute value of the asymmetry determine the direction and degree
of $\gamma'$ polarization. We have analyzed the process in a wide range of
laser intensity that covers existing and future experiments. Our results
provide additional knowledge for studying nonlinear multi-photon effects in
quantum electrodynamics and can be used in planning experiments at envisaged
laser facilities.
