  Binary black hole spins are among the key observables for gravitational wave
astronomy. Among the spin parameters, their orientations within the orbital
plane, $\phi_1$, $\phi_2$ and $\Delta \phi=\phi_1-\phi_2$, are critical for
understanding the prevalence of the spin-orbit resonances and merger recoils in
binary black holes. Unfortunately, these angles are particularly hard to
measure using current detectors, LIGO and Virgo. Because the spin directions
are not constant for precessing binaries, the traditional approach is to
measure the spin components at some reference stage in the waveform evolution,
typically the point at which the frequency of the detected signal reaches 20
Hz. However, we find that this is a poor choice for the orbital-plane spin
angle measurements. Instead, we propose measuring the spins at a fixed
dimensionless time or frequency near the merger. This leads to significantly
improved measurements for $\phi_1$ and $\phi_2$ for several gravitational wave
events. Furthermore, using numerical relativity injections, we demonstrate that
$\Delta \phi$ will also be better measured near the merger for louder signals
expected in the future. Finally, we show that numerical relativity surrogate
models are key for reliably measuring the orbital-plane spin orientations, even
at moderate signal-to-noise ratios like $\sim 30-45$.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology