Lindsey's response is so good I almost didn't have anything to add. I'll say though that the most important thing is communication. Ideally you'd arrange things beforehand, but I can only speak about the surprise situation, where the band can't put down their instruments, which has happened to me.

Listen to the music and know where the phrases are. You'll want to make sure the band knows when you're starting for reals. Whatever you do, make it really obvious. So make eye contact, gesticulate, and get an acknowledgement just before you begin. You could break the illusion a little by saying, "the dance starts here with..." but I wouldn't consider that enough by itself to cue the band. Even more illusion-breaking: at the appropriate place in the music, you could even give the band potatoes: "5, 6, balance now".

It's not a big deal though if the first swing happens to be 20 counts long or something. Tell the dancers to keep swinging, etc. I find it's easier to adjust the dancers to fit the music.

Now of course if you really want to break it up, you can tell the band to knock it off, perhaps if you're teaching something intricate and need the dancers' full attention. It's a bit of a letdown but maybe better than crashing later. Be apologetic.

I think dances that might work especially well, for when you want to plan a rolling start in advance, are those that have an unusual starting position that is slightly disorienting to go back to, so that it's easier to just carry on. I'm thinking of Chuck the Budgie and other such Rick Mohr dances.

Yoyo Zhou