Quote:
So in essence, the ganged breaker just knows how to give the right phase to each, or are there 2 separate bars back there for each phase?


Yes, the bus bars in the breaker box alternate phases in adjacent slots, so if you plug in a 220V breaker you get one phase on each side. Electrically you don't need to use a 220V breaker, you could plug in two 15 amp breakers adjacent to one another an it would be perfectly safe. However, I believe that code requires the ganged breaker.

Quote:
A colleague pointed out the existence of mini tandem breakers, meaning I can get 2 separate 15 A circuits in the space of one traditional breaker.


I thought about these after my first post. If you are out of space in your breaker box the tandem breakers are handy. Just remember that the two sides on a tandem breaker are on the same phase, so the two circuits on the tandem should not share the neutral conductor.

-Jeff