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Binary black holes with misaligned spins will generically induce both precession and nutation of the orbital angular momentum 𝐋 about the total angular momentum 𝐉. These phenomena modulate the phase and amplitude of the gravitational waves emitted as the binary inspirals to merger. We introduce a “taxonomy” of binary black-hole spin precession that encompasses all the known phenomenology, then present five new phenomenological parameters that describe generic precession and constitute potential building blocks for future gravitational waveform models. These are the precession amplitude ⟨𝜃𝐿⟩, the precession frequency ⟨Ω𝐿⟩, the nutation amplitude Δ⁢𝜃𝐿, the nutation frequency 𝜔, and the precession-frequency variation Δ⁢Ω𝐿. We investigate the evolution of these five parameters during the inspiral and explore their statistical properties for sources with isotropic spins. In particular, we find that nutation of 𝐋 is most prominent for binaries with high spins (𝜒≳0.5) and moderate mass ratios (𝑞∼0.6).