If they are alternating like you said, it sounds like cross wires. I am confused at how it might be a ground if both headlights have high and low (dim) beams. If they have both, yet opposite times (one dim and one high) each time you hit the switch, wouldn’t one light have to be wired backwards?

I am not a smart man, but I accidentally did this when I rewired my truck.

Chris

If I am wrong, please explain so I have a better understanding.