We construct a non-linear theory of interacting spin-2 fields that is
invariant under the partially massless (PM) symmetry to all orders. This theory
is based on the SO(1,5) group, in analogy with the SO(2,4) formulation of
conformal gravity, but has a quadratic spectrum free of ghost instabilities.
The action contains a vector field associated to a local SO(2) symmetry which
is manifest in the vielbein formulation of the theory. We show that, in a
perturbative expansion, the SO(2) symmetry transmutes into the PM
transformations of a massive spin-2 field. In this context, the vector field is
crucial to circumvent earlier obstructions to an order-by-order construction of
PM symmetry. Although the non-linear theory lacks enough first class
constraints to remove all helicity-0 modes from the spectrum, the PM
transformations survive to all orders. The absence of ghosts and strong
coupling effects at the non-linear level are not addressed here.