I'm reading various introductions to SPDC, and they all seem to take the state of the entangled photons to be the Bell state that is asymmetric under exchange: $$\left(\left| HV\right>\left| VH\right>-\left| VH\right>\left| HV\right>\right)\frac{1}{\sqrt2}$$$$\Big( \left| H\right>_1\left| V\right>_2-\left| V\right>_1\left| H\right>_2 \Big)\frac{1}{\sqrt2}$$ but I haven't run across anyone explaining why. I wouldn't expect the asymmetric case for photons (bosons).