I was using pieces from CDROM heads for beam splitters. After a bit of googling, I've found some of them are actually polarizing beam splitters, which I was not expecting. While they are potentially very useful stuff, in my case their polarizing properties can be the reason I can't get any positive results.

UPDATE. After a bit of mucking around with polarizers, I figured out that the mirror is not at fault. While the transmitted light is indeed very polarized, the reflected one is not, I didn't notice even a tiny bit of polarization. So, while the mirror is not optimal, it is far from being the culprit.