We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π⁰₃ iff the invariants are countable sets of reals, it is potentially Π⁰₄ iff the invariants are countable sets of countable sets of reals, and so on. The proofs use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models.