For a swap which has been entered into by two parties (one paying floating and the other paying on equity index returns), what would be the value of only the equity side payments on the LIBOR reset date??

I think it would be 0 for both sides.

This confuses me as well, how when we’re valuing a fixed-rate payer/equity return receiver swap at a point in time, you just value the swap as if the equity notional resets at each date and assign it a 0-value. Doesn’t make sense to me but I’m sure I’m missing something.

Think of it this way.

The floating rate gets reset on each payment date, so does equity. So changes in rate(from initiation doesnt affect the value)

Fixed rate, you have fixed payment. Based on changes in rate your PV of the future payment changes.

The payer of equity returns would have a value equal to the face value of the floating bond minus any gains to the index on LIBOR reset. But note that usually, equity swaps in this case would include a spread on the floating payment, because equity returns are not risk free, and you should discount both payments at the same rate.

If the index did not move since swap initiation, it would be zero (in case of no spread).