Maybe I'm missing something, but it seems to me this phenomenon is entirely explainable classically (and fairly simply) by using the fact that linear filters dot product the polarisation vector of the incoming light with the polarisation vector of the filter. A dot product of orthogonal vectors gives you 0, while a dot product of vectors rotated 45° gives you a non-zero answer (meaning if you do it twice you still get a non-zero answer). We did a fairly simple proof of this in second-year undergraduate physics. It's only surprising if you make certain assumptions about linear filters that are not correct.
Of course you need a quantum explanation if you're treating light as photons, but quantum mechanics is not needed to explain this phenomenon (unlike other phenomena).