  We construct a class of extended shift symmetries for fields of all integer
spins in de Sitter (dS) and anti-de Sitter (AdS) space. These generalize the
shift symmetry, galileon symmetry, and special galileon symmetry of massless
scalars in flat space to all symmetric tensor fields in (A)dS space. These
symmetries are parametrized by generalized Killing tensors and exist for fields
with particular discrete masses corresponding to the longitudinal modes of
massive fields in partially massless limits. We construct interactions for
scalars that preserve these shift symmetries, including an extension of the
special galileon to (A)dS space, and discuss possible generalizations to
interacting massive higher-spin particles.
