I suspect you'll find that the pairs of diodes are connected with reverse polarity, relying on the reverse breakdown voltage being greater than the forward voltage to avoid damaging the LED which is powered in reverse. Hence half the LEDs could be controlled active low with the common rail held high, and the other half would be powered active high with the common rail being held low. By modulating each LED fast enough, persistence of vision would make it appear to be a constant brightness. I don't see any separate current-limiting resisters for the LEDs, which is odd.