Gauge invariance of the linearized Riemann tensor

Under $h_{\mu \nu }\mapsto h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, every contribution to the linearized Riemann tensor contains three commuting derivatives of $\xi$ and cancels pairwise. Linearized curvature is therefore gauge invariant about flat spacetime.