There is a LOT of confusion on how to implement quad breakers, let me see if I can clear it up.
A) Any 2 POLE breaker that is used for a 120/240V load or an MWBC MUST HAVE A COMMON TRIP. In Quad breakers that is ONLY the middle two poles, it is NEVER the outer two poles, there is no way to make them have a common trip from afar (without joining them with the inner poles, see C)).
B) The outer two poles can be joined together by a clip, and THAT can be used for STRAIGHT 240V loads, meaning appliances or circuits that do NOT have a neutral run to them, such as a water heater. That is the ONLY legitimate use for the outer pole clips.
C) (I’m not 100% sure on this, but this is what I was told by a Schneider guy) Schneider (Square D Homeline) has that quad that has ALL FOUR handles tied together. In that case, the outer two poles ARE common trip, but they are ALSO common trip with the INNER two poles as well, because that’s the only way to make that work. So the drawback to this is that if one appliance overloads, you will shut down BOTH appliances. A legit application for that might be a tankless water heater or stove and oven combo where each element needs its own 120/240 circuit, but if one trips, they both trip. You can use it any way you like of course, but you have to accept that quirk of both separate circuits tripping together.