When done properly, the two breakers would be tied together, so either both
would be on or off.

Good point. I did read something about a double pole breaker. That IS
what I should have in my box. So either I don't have one or the
neutrals that I was disconnecting actually weren't part of my intended
circuit at all. There were a lot of wires in that box, perhaps another
group was actually the neutral for my circuit. The wiring is 50 years
old and it's hard to tell what color these wires were supposed to be!