CiviEvent, group by relationship
CiviMobile (new version) works nicely for checking people into an event, but I think it's going to do its own sorting regardless of anything you display in the normal Civi GUI.

If you're on Drupal then a view using the DancePartner relationship with a VBO might do what you need to display the partners and register attendance.

Do you want to register and check in individuals or couples?  

If you are more interested in couples, then you could potentially repurpose/abuse 'household' as 'dance partnership' since you can register households as event participants.  You could then use CiviMobile and check in the 'Marie &amp; Peter partnership' household.  You would need to check the rest of your workflow but it fits the objective of checking in a couple via CiviMobile.