Usually everything in a contra dance matches up exactly with the music. The tunes repeat every 64 beats, and are divided into two 32-beat halves (A and B), each of which is broken up into two 16-beat halves (A1 and A2, B1 and B2), each of which is broken into two 8-beat halves (not usually named). Each dance is choreographed so the transitions between moves match up with transitions between sections of music. For example:
The Baby Rose by David Kaynor
A1 (16) Neighbor balance and swing A2 (8) Circle left 3/4 (8) Partner dosido B1 (16) Partner balance and swing B2 (8) Robins chain (8) Star left
This is saying that the first 16 beats of the dance (A1) are "neighbor balance and swing". The next 16 beats (A2) are divided into 8 beats for a "circle left 3/4", and then another 8 for a "partner dosido". The dancers will adjust how quickly they're moving through the figures to make the moves line up exactly, and this tight timing makes the dance feel very satisfying. Music that clearly communicates these divisions, either by the tune you've chosen or how you're playing it, is "well phrased": the phrases the dancers hear are the ones needed to support the dance (more).
There are a few places where there's extra time and it isn't a problem to start the next figure a bit early. For example, consider a common combination:
(8) Circle left 3/4 (8) Swing
It usually doesn't take the full eight beats to circle 3/4, and the swing is a move that can take a range of lengths, so it's more common to see people dance this as:
(6) Circle left 3/4 (10) Swing
This minor departure from power-of-two divisions is about as far as the typical dance will get from this pattern. A few years ago, however, I wrote about a few dances that have entirely unphrased B-parts, and last night I noticed we were dancing to another. Writing the dance as if it had used conventional timing it would be:
Never Too Late
by Tom Hinds
A1 (8) Larks dosido (8) Larks allemande right 1.5 A2 (16) Partner balance and swing B1 (8) Right and left through (8) Robins chain B2 (8) Robins right shoulder round (8) Neighbor swing
The B part doesn't really dance like that, though. I counted how it was going for us, and it was pretty close to:
B (6) Right and left through (6) Robins chain (6) Robins right shoulder round (14) Neighbor swing
With a less energetic crowd I could imagine a division more like:
B (8) Right and left through (6) Robins chain (6) Robins right shoulder round (10) Neighbor swing
Even within a single hall, some minor sets ("hands fours") will be dancing with a range of timings, all depending on how quickly they happen to end up going through the figures.
Looking at this from the perspective of a musician, it's very tricky! Normally you want to be making the phrase of the music very clear so the dancers don't get lost. If they don't know when to start and stop things the dance will fall apart. In this dance, however, with different parts of the hall using slightly different timing, that's not going to work. And many of the dancers will be putting their transitions in places that don't match any phrase you can emphasize. Instead, you'd want to play in a way that highlights only three transitions:
Music with no clear phrasing won't work, because you do need to make those three transitions clear, but you don't want music that boldly indicates the end of a phrase halfway through the A2, or anywhere during the B. You can go looking for a tune that has this exact phrasing, but I don't think you're going to find one. This is a case where you would want to find a tune that's pretty close, and then adapt how you play it for this specific dance.