We report spectropolarimetric observations of the quasar E1821+643 (z = 0.297), which suggest that it may be an example of gravitational recoil due to anisotropic emission of gravitational waves following the merger of a supermassive black hole (SMBH) binary. In total flux, the broad Balmer lines are redshifted by approximate to 1000 km s(-1) relative to the narrow lines and have highly red asymmetric profiles, whereas in polarized flux the broad Ha line exhibits a blueshift of similar magnitude and a strong blue asymmetry. We show that these observations are consistent with a scattering model in which the broad-line region has two components, moving with different bulk velocities away from the observer and toward a scattering region at rest in the host galaxy. If the high-velocity system is identified as gas bound to the SMBH, this implies that the SMBH is itself moving with a velocity similar to 2100 km s(-1) relative to the host galaxy. We discuss some implications of the recoil hypothesis and also briefly consider whether our observations can be explained in terms of scattering of broad-line emission originating from the active component of an SMBH binary, or from an outflowing wind.