  Gravitational waves from precessing black-hole binaries exhibit features that
are absent in nonprecessing systems. The most prominent of these is a
parity-violating asymmetry that beams energy and linear momentum preferentially
along or opposite to the orbital angular momentum, leading to recoil of the
binary. The asymmetry will appear as amplitude and phase modulations at the
orbital frequency. For strongly precessing systems, it accounts for at least 3%
amplitude modulation for binaries in the sensitivity band of ground-based
gravitational-wave detectors, and can exceed 50% for massive systems. Such
asymmetric features are also clearly visible when the waves are decomposed into
modes of spin-weighted spherical harmonics, and are inherent in the waves
themselves---rather than resulting from residual eccentricity in numerical
simulations, or from mode-mixing due to precession. In particular, there is
generically no instantaneous frame for which the mode decomposition will have
any symmetry. We introduce a method to simplify the expressions for waveforms
given in analytical relativity, which can be used to combine existing
high-order waveforms for nonprecessing systems with expressions for the
precessing contributions, leading to improved accuracy and a unified treatment
of precessing and nonprecessing binaries. Using this method, it is possible to
clarify the nature and the origins of the asymmetries and show the effects of
asymmetry on recoils more clearly. We present post-Newtonian (PN) expressions
for the waveform modes that include these terms, complete to the relative 2PN
level in spin (proportional to $v^4/c^4$ times a certain combination of the
spins). Comparing the results of those expressions to numerical results, we
find good qualitative agreement. We also demonstrate how these expressions can
be used to efficiently calculate waveforms for gravitational-wave astronomy.
