See it all the time, especially with teachers--husbands and wives--lovers.

How is it properly handled socially when you both still dance at the same scenes (assuming that things didn't get too bitter)?

I have noticed that usually one couple drops out of dancing, moves, or relocate your dancing if your city is big enough.

In a regular relationship, unless you have kids, you don't see the person that much and you just move on.

But with dancing there they are on they floor with a new partner or friend and you are with a new partner or friend--and it just feels uncomfortable.
Lots of dancers use dancing as their main social outlet or lifestyle and it is a big inconvenience to change studios, teachers, friends and etc.

The loss is not so much the person you were dancing with, but the dancing that you created together.

Even if you're really ticked off, it's hard to ignore that you probably had some good times/accomplishments together too. Being adult is being able to recognize those most of the time, and getting/looking away in the moments when the negative memories are just too strong. Ex partners, at least those who have continued working on their dancing, are some of my best friends.