I don't remember seeing many handle ties in your panel. MWBCs require those now. They do not require common trip, however. That means Eaton breakers like BD220220 (not BDC220220) are fair game.
Left-right handed slimlines are dead as a product and will never return. That bus clip is quite "dainty" and also poorly guarded (being on the edge of the breaker, rather than center like GE thins) That made them prone to damage, and if clamping force is not correct it'll burn up the bus stab and put 4 breaker positions out of action. Or worse.
Tandems and quadplex are simply slimlines factory bolted together, or a little bit more in one case. Quadplex breakers come in two major formats.
- 2-pole breaker in the middle and two singles on the outside.
- 2-pole breaker in the middle and another 2-pole breaker on the outside, with a hinky looking handle tie tying them together.
The hinky looking ties make you think "how can that possibly cause common trip? It will simply tram." Sure but here's the breaker secret inside a secret. Breakers "trip free" - they trip even if the handle is held or locked "on". Therefore... handle-ties would not work as a common-trip method, and are not used for that. Common trip is a separate internal mechanism inside the breaker. The handle tie is to help maintainers shut off the whole circuit so they don't get nailed.
...and that's why... on Eaton breakers the common-trip mechanism might not be there. They make quads in two flavors: with and without.
"Without" is acceptable for MWBCs if they do not serve any 120V loads. "Without" is also acceptable for 240V-only loads which do not have a neutral. I don't why Eaton bothers to produce no-common-trip quads. Nobody ever died because their breaker was common-trip that didn't need to be. Most makers do common trip on the inner, and not on the outer.
With MWBCs, you need to wire then like a 120/240V load with the hots on opposite pole. So you don't use them with tandems. On a quadplex you put a given MWBC on the inner breaker or the outer one.
It helps to be able to visualize your panel's spaces as full spaces, in which two slimlines share a space. And then see the red/black striping a-la my answer on tandems.