  Remnants of binary black-hole mergers can gain significant recoil or kick
velocities when the binaries are asymmetric. The kick is the consequence of
anisotropic emission of gravitational waves, which may leave a characteristic
imprint in the observed signal. So far, only one gravitational-wave event
supports a non-zero kick velocity: GW200129_065458. This signal is also the
first to show evidence for spin-precession. For most other gravitational-wave
observations, spin orientations are poorly constrained as this would require
large signal-to-noise ratios, unequal mass ratios or inclined systems. Here we
investigate whether the imprint of the kick can help to extract more
information about the spins. We perform an injection and recovery study
comparing binary black-hole signals with significantly different kick
magnitudes, but the same spin magnitudes and spin tilts. To exclude the impact
of higher signal harmonics in parameter estimation, we focus on equal-mass
binaries that are oriented face-on. We generate signals with PhenomXO4a, which
includes mode asymmetries. These asymmetries are the main cause for the kick in
precessing binaries. For comparison with an equivalent model without
asymmetries, we repeat the same injections with PhenomXPHM. We find that
signals with large kicks necessarily include large asymmetries, and these give
more structure to the signal, leading to more informative measurements of the
spins and mass ratio. Our results also complement previous findings that argued
precession in equal-mass, face-on or face-away binaries is nearly impossible to
identify. In contrast, we find that in the presence of a remnant kick, even
those signals become more informative and allow determining precession with
signal-to-noise ratios observable already by current gravitational-wave
detectors.
