This is pretty much what goes on at ice dance weekends. My partner and I are planning to attend two this summer. Skaters need not have a partner to attend, though. In fact, it's considered bad form to dance with your partner the whole time. The rotation is much more informal than you describe, but we are expected to change partners for each dance.
When you register, you indicate your level (i.e. which dances you know). The organizers then prepare a program of dances that covers the abilities of the dancers attending. This is posted in the rink, so you can see the order of dances. Usually they try to alternate lower and higher level dances so nobody has to sit out too long. If you don't know a dance, you just hang by the boards or step into the hockey box for that one. If someone asks you to do a dance you don't know, you politely decline, explaining that you don't know that one yet. If you know the steps, but not well enough to safely partner them yet, you can do the dance solo at the end of the line after all the couples. There are often more ladies than men, so each dance is done twice to give more ladies a chance to skate. Ladies who don't have a partner for a particular dance may also solo it at the end of the line if they want.
Sometimes we do "fun" activities like mixers much as you described - the men and ladies form concentric circles and skate around in opposite directions. When the music stops, you partner with the person opposite you for the next dance. There are a few "non-test" dances, like the Golden Skaters Waltz, that get done at social dance weekends, and sometimes we do variations, like skating a dance the opposite direction. Some weekends include clinics, and most also include off-ice social activities. There's always food!
Some places are lucky enough to have social dance sessions every week - I'm thinking of the Starlight Ice Dance Club in Minnesota - but most of us travel to ice dance weekends periodically because we don't have enough dancers at our rinks to support dedicated social dance sessions. There's always a social dance on Sunday morning at Adult Nationals. That was my first social dance experience - I had just started dance and could only partner the first few, but I still had a good time.
Oh - forgot to mention that for dance weekends you can usually register for all or part of it, so you can still participate even if you can't make the whole thing. Obviously, if you had to travel any distance to get there, you'd want to stay for the entire weekend.
Apologies if you already know all this, but there might be others who have wondered about what happens at ice dance weekends.