By "Only once," I had intended to reinforce my contention in the first sentence in the post. The AR1 does not make two changes to polarity. It makes one, only one, at whichever end has the conflict. That, in turn, depends on which end the train enters the controlled/switched loop.
Logically, if there is a conflict in phase/polarity when the train enters at the bottom of the loop in the correctly oriented diagramme, it will NOT have to alter the phase/polarity when the train exits a few seconds later at the far gap. That's because the rails across the gap are in phase/polarity there.
Visualize this yourself imagining a train entering at either end of the reversing section, imagine that there's a conflict, or not, and then imagine what must take place (or not) at the other end. You'll see that the AR only has to make a single "correction", not one at both ends.
It is for this reason that the AR must only control what lies between the two gaps, and only that length of 'reversible' track. Meanwhile, out on the greater layout, all wires must be the same on all rails, and they must get power from the DCC system only. So does the AR...it is fed power directly from the DCC outputs. The AR's outputs control ONLY the length of tracks between the gaps, and it matters not which wire/output terminal powers which rail because no matter which way you phase the rails, one end is going to conflict.
Does that all make sense?