Gravitational waves (GW), as light, are gravitationally lensed by intervening matter, deflecting their trajectories, delaying their arrival and occasionally producing multiple images. In theories beyond general relativity, new gravitational degrees of freedom add an extra layer of complexity and richness to GW lensing. We develop a formalism to compute GW propagation beyond general relativity over general space-times, including kinetic interactions with new fields. Our framework relies on identifying the dynamical propagation eigenstates (linear combinations of the metric and additional fields) at leading order in a short-wave expansion. We determine these eigenstates and the conditions under which they acquire a different propagation speed around a lens. Differences in speed between eigenstates cause birefringence phenomena, including time delays between the metric polarizations (orthogonal superpositions of $h_+$, $h_×$) observable without an electromagnetic counterpart. In particular, GW echoes are produced when the accumulated delay is larger than the signal's duration, while shorter time delays produce a scrambling of the waveform. We also describe the formation of GW shadows as nonpropagating metric components are sourced by the background of the additional fields around the lens. As an example, we apply our methodology to quartic Horndeski theories with Vainshtein screening and show that birefringence effects probe a region of the parameter space complementary to the constraints from the multimessenger event GW170817. In the future, identified strongly lensed GWs and binary black holes merging near dense environments, such as active galactic nuclei, will fulfill the potential of these novel tests of gravity.