  In this paper we derive a differential identity for linearized gravity on the
Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We
show that a linear combination of second derivatives of the linearized Weyl
tensor can be formed into a complex symmetric 2-tensor $\mathcal{M}_{ab}$ which
solves the linearized Einstein equations. The identity makes this manifest by
relating $\mathcal{M}_{ab}$ to two terms solving the linearized Einstein
equations by construction. The self-dual Weyl curvature of $\mathcal{M}_{ab}$
gives a covariant version of the Teukolsky-Starobinsky identities for
linearized gravity which, in addition to the two classical identities for
linearized Weyl scalars with extreme spin weights, includes three additional
equations. In particular, they are not consequences of the classical
Teukolsky-Starobinsky identities, but are additional integrability conditions
for linearized gravity. The result has direct application in the construction
of symmetry operators and also yields a set of non-trivial gauge invariants for
linearized gravity.
