The present article deals with gravitational-wave propagation affected by higher-derivative terms that break spacetime symmetries. Two dimension-6 contributions of the gravitational Standard Model Extension pose our starting point. A linearization of the action implies a wave equation that contains additional terms with spacetime-constant background fields. The modified dispersion relations in covariant form for gravitational waves are derived, where we distinguish between non-birefringent and birefringent sectors. A classification of the coefficients in terms of sets with index structures resembling those of the electromagnetic fields has proven to be valuable. We constrain the nonbirefringent coefficients based on the measured arrival time difference between the gravitational wave and photons from the events GW170817 and GRB 170817A, respectively. Bounds on the birefringent coefficients result from the absence of a perceivable separation of the two modes in the event GW150914. These findings quantify the extent to which standard linearized gravity is valid based on the modifications considered.