  The spectroscopic study of black hole quasinormal modes in gravitational-wave
ringdown observations is hindered by our ignorance of which modes should
dominate astrophysical signals for different binary configurations, limiting
tests of general relativity and astrophysics. In this work, we present a
description of the quasinormal modes that are excited in the ringdowns of
comparable mass, quasi-circular precessing binary black hole coalescences -- a
key region of parameter space that has yet to be fully explored within the
framework of black hole spectroscopy. We suggest that the remnant perturbation
for precessing and non-precessing systems is approximately the same up to a
rotation, which implies that the relative amplitudes of the quasinormal modes
in both systems are also related by a rotation. We present evidence for this by
analyzing an extensive catalog of numerical relativity simulations. Additional
structure in the amplitudes is connected to the system's kick velocity and
other asymmetries in the orbital dynamics. We find that the ringdowns of
precessing systems need not be dominated by the ${(\ell,m)=(2,\pm 2)}$
quasinormal modes, and that instead the $(2,\pm 1)$~or~$(2,0)$ quasinormal
modes can dominate. Our results are consistent with a ringdown analysis of the
LIGO-Virgo gravitational wave signal GW190521, and may also help in
understanding phenomenological inspiral-merger-ringdown waveform model
systematics.
