Scattering of an unpolarized photon off an electron

Hi all!

I have been told that if an unpolarized photon hits an electron in a Thomson scattering the outcoming photon will be polarized because of the electron's spin. I didn't understand what it means, nor do I get how an electron reacts to an unpolarized photon: in Classical Electrodynamics, if an electron is hit by an electromagnetic wave, it starts to vibrate in the direction of the electric field, i.e. of polarization. If the photon is unpolarized, there is no preferred direction for the electron to vibrate in, so I do not understand how it can even emit an outcoming electromagnetic wave.

Thanks in advance!