A glass mirror (with metal backing layer) will reverse the polarisation of circularly polarised light upon reflection.
A polished piece of metal will also reverse the polarisation of circularly polarised light upon reflection. (I have tested and confirmed this for myself).
wikipedia states the reason a mirror will reverse the polarisation of circularly polarised light is:
…[A]s a result of the interaction of the electromagnetic field with the conducting surface of the mirror, both orthogonal components are effectively shifted by one half of a wavelength.
However, my understanding of mirrors is that only a polished piece of metal will phase shift a wavelength by half a wavelength, whereas a glass mirror (with metal backing layer) will not produce a phase shift. For example wikipedia which states:
According to Fresnel equations there is only a phase shift if n2 > n1 (n = refractive index). This is the case in the transition of air to reflector, but not from glass to reflector
Best Answer
While rob is correct about the quantum mechanical picture I think that this case is at least as easy to understand as in the classical description.
Classically circular polarization can be described in terms of a time-varying linear polarization, so let's just look at two points on a wave.
I'm going to chose a beam in the $+z$ direction to examine two points on the wave: one where the polarization currently points along $+\hat{x} - \epsilon \hat{y}$, and a very short time later where the polarization is in the $+\hat{x} + \epsilon \hat{y}$ direction. The wave has right-handed circular polarization.
Now we let the beam bounce off a mirror in the $x\text{--}y$ plane. This reverses the direction of propagation but leaves the time-order in which are two points of interest pass any given point unaffected. A little thought suffices to show that the reflected wave has left-handed circular polarization.