  In linearized gravity, two linearized metrics are considered
gauge-equivalent, $h_{ab} \sim h_{ab} + K_{ab}[v]$, when they differ by the
image of the Killing operator, $K_{ab}[v] = \nabla_a v_b + \nabla_b v_a$. A
universal (or complete) compatibility operator for $K$ is a differential
operator $K_1$ such that $K_1 \circ K = 0$ and any other operator annihilating
$K$ must factor through $K_1$. The components of $K_1$ can be interpreted as a
complete (or generating) set of local gauge-invariant observables in linearized
gravity. By appealing to known results in the formal theory of overdetermined
PDEs and basic notions from homological algebra, we solve the problem of
constructing the Killing compatibility operator $K_1$ on an arbitrary
background geometry, as well as of extending it to a full compatibility complex
$K_i$ ($i\ge 1$), meaning that for each $K_i$ the operator $K_{i+1}$ is its
universal compatibility operator. Our solution is practical enough that we
apply it explicitly in two examples, giving the first construction of full
compatibility complexes for the Killing operator on these geometries. The first
example consists of the cosmological FLRW spacetimes, in any dimension. The
second consists of a generalization of the Schwarzschild-Tangherlini black hole
spacetimes, also in any dimension. The generalization allows an arbitrary
cosmological constant and the replacement of spherical symmetry by planar or
pseudo-spherical symmetry.
