Ah, so. So when those folks were saying "The currents through the capacitor in each direction must be equal", what they really meant is that in order for the LED to light in the forward direction, some current must flow through the capacitor in the reverse direction in order to discharge the capacitor: would you agree?
And I suppose that if you look at the current over time (di/dt), then the currents would have to be equal.
Just an editorial note: in case you don't see what I'm driving at, it's not sufficient to make an assertion ("the currents must be equal") without some explanation, even if it makes perfect sense to you. (Which is also a roundabout way of thanking The Electrician and Diver300 for their patience.)
I think that we have come to an agreement.
You have a slight mathematical error. The current over time is ∫i dt, which has to be zero on a capacitor (give or take leakage current) while di/dt is the rate of change of current, which could be anything.