  We study interacting massive spin-1 theories in de Sitter (dS) and anti-de
Sitter (AdS) space that possess shift symmetries parametrized by (A)dS Killing
vectors. We show how they emerge from the massless limit of massive spin-2
theories on (A)dS space. In the case of massive gravity, the corresponding
spin-1 theory realizes a symmetry breaking pattern that takes two copies of the
(A)dS isometry group down to a diagonal subgroup. By taking the flat space
limit of this theory, we find a new symmetry of the decoupling limit of massive
gravity in flat space. This symmetry acts on the vector modes, is parametrized
by an antisymmetric tensor, and fixes the nonlinear structure of the
scalar-vector sector of the decoupling limit.
