In the Einstein-Podolsky-Rosen experiment, the polarisations of the photons are not independent of each other. If the filters with which one determines the polarisation of the photons are perpendicular to each other, then when a photon is registered at one detector, another one is also always found at the the other.
This means that the polarisations of both photons are always perpendicular to each other. However, according to quantum mechanics, before the measurement the photons have no definite polarisation whatsoever!. The fact that still the polarisations are dependent on each other is called entanglement.