What are some rare differentials that might cause constant vertigo in a person?

Dizzying...: Well, if you want rare etiologies of constant vertigo then my answer is brainstem tumors, cranial nerve 8 tumors, and undiagnosed scar tissue from prior penetrating wound/prior surgery.

See a subspecialist: If you are asking for a rare differential... It's likely that you have already been extensively evaluated. I would see an ENT who specializes in ear diseases: neurotologist or a neurologist who likes to see dizzy patients.