I got a new quad breaker (Murray) for two 240V circuits. I noticed that the metal bar that connects the two on the outside ( that is, out of X X X X, I mean the X - - X circuit) works great for turning both on at the same time, but I can flip one of them off and the bar doesn't pull the other one down. Is something horribly wrong, or are these breakers designed with some kind of internal linkage that doesn't depend on the physical switch part to trip open the second one if the first one trips?

Weaker half of breaker will trip when a leg hits ground/overloads. Even with an equal load, 1/2 of breaker may not trip at same, due to half being weaker, not exact, The half that trips handles the fault.
But to work on it safe, you would verify both halfs are switched off.