How can two different breakers both turn off the same circuit?
That could be a split bus panel, meaning that there are two sets of power distribution buses.  It used to be allowed by something called the "6 handle rule" (which allowed up to six of them in a service panel) back when circuit breakers cost a small fortune.

They were wired like this:

And might look like this: 
Note the gap in the breaker buses at center where there are no breakers, but you can see heavy black wires connecting to the right end of the secondary buses.

The cables from the meter connect at right (which would conventionally be the top of the panel) and feeds the six double breakers.  Five of them feed directly to heavy loads:  range, water heater, dryer, baseboard heaters, etc.  The top-left double ganged breaker feeds the second bus in the left half of the photo.

In this configuration, any circuit fed from the lower bus would be shut off if either its own breaker were switched off, or the feed breaker&mdash;which also shuts off almost everything in the building.