After ortho residency, an orthodontist can either join up with an existing "practice" (private practice, academia, or institutional) or start up on their own, whatever they feel like doing.
The only caveat is that if you call yourself a specialist in orthodontics, then you can't do any general dentistry on the side, whereas even if you graduated from an ortho residency, but don't calll yourself an ortho specialist, then you can work as a general dentist and do ortho too.