  Extensions to General Relativity (GR) allow the polarization of gravitational
waves (GW) from astrophysical sources to suffer from amplitude and velocity
birefringence, which respectively induce changes in the ellipticity and
orientation of the polarization tensor. We introduce a multi-messenger approach
to test this polarization behavior of GWs during their cosmological propagation
using binary sources, for which the initial polarization is determined by the
inclination and orientation angles of the orbital angular momentum vector with
respect to the line of sight. In particular, we use spatially-resolved radio
imaging of the jet from a binary neutron star (BNS) merger to constrain the
orientation angle and hence the emitted polarization orientation of the GW
signal at the site of the merger, and compare to that observed on Earth by GW
detectors. For GW170817 we constrain the deviation from GR due to amplitude
birefringence to $\kappa_A = -0.12^{+0.60}_{-0.61}$, while the velocity
birefringence parameter $\kappa_V$ remains unconstrained. The inability to
constrain $\kappa_V$ is due to the fact that Virgo did not detect GW170817, and
measurements of the polarization orientation require information from a
combination of multiple detectors with different alignments. For this reason,
we also mock future BNS mergers with resolved afterglow proper motion and
project that $\kappa_V$ could be constrained to a precision of $5\,$rad
(corresponding to an angular shift of the GW polarization of
$\delta\phi_V\approx 0.2\,$rad for a BNS at $100\,$Mpc) by a future network of
third-generation ground-based GW detectors such as Cosmic Explorer and the
radio High Sensitivity Array. Crucially, this velocity birefringence effect
cannot be constrained with dark binary mergers as it requires polarization
information at the emission time, which can be provided only by electromagnetic
emission.
