A complete measurement of a black-hole recoil through higher-order gravitational-wave modes

General relativity predicts that gravitational waves (GWs) carry linear momentum. Consequently, the remnant black hole of a black-hole merger can inherit a recoil velocity or ``kick'' of crucial implications in, e.g., black-hole formation scenarios. While the kick magnitude is determined by the mass ratio and spins of the source, estimating its direction requires a measurement of the \textit{two orientation angles} of the source. While the orbital inclination angle is commonly reported in GW observations, the scientific potential of the azimuthal one has not been exploited to date. We show how the presence of more than one GW emission mode allows one to constrain this angle and, consequently, the kick direction of a real GW event. We analyse the GW190412 signal, which contains higher-order modes, with a numerical-relativity surrogate waveform model for black-hole mergers. We rule out kick magnitudes below the typical escape velocity of dense globular clusters $v_{\text{esc}}\approx 50$\,km/s with a Bayes Factor of $\simeq 21$ (or $\simeq 95\%$ probability). The kick forms angles $\theta_{KL}^{-100M}=32^{+35}_{-14}\,\deg$ with the orbital angular momentum defined at a reference time $t_{\rm ref}=-100\,M$ before merger (with $M$ denoting the system mass in geometric units), $\theta_{KN}=44^{+19}_{-17}\,\deg$ with the line-of-sight. The projections of the kick and line-of-sight onto the orbital plane form an angle $\phi_{KN}^{-100M}=69^{+33}_{-38}\,\deg$. All quantities are quoted at a $90\%$ credible level. Finally, by analyzing numerically simulated signals, we show that recoils can be estimated in an unbiased way using the NRSur7dq4 waveform model. We briefly discuss the potential application of this type of measurement for multi-messenger observations of black-hole mergers occurring in Active Galactic Nuclei.