  Orbital eccentricity and spin precession are precious observables to infer
the formation history of binary black holes with gravitational-wave data. We
present a post-Newtonian, multi-timescale analysis of the binary dynamics able
to capture both precession and eccentricity over long inspirals. We show that
the evolution of an eccentric binary can be reduced that of effective source on
quasi-circular orbits, coupled to a post-Newtonian prescription for the secular
evolution of the eccentricity. Our findings unveil an interplay between
precession and eccentricity: the spins of eccentric binaries precess on shorter
timescales and their nutation amplitude is altered compared to black holes on
quasi-circular orbits, consequently affecting the so-called spin morphology.
Even if binaries circularize by the time they enter the sensitivity window of
our detectors, their spin orientations retain some memory of the past evolution
on eccentric orbits, thus providing a new link between gravitational-wave
detection and astrophysical formation. At the same time, we point out that
residual eccentricity should be considered a source of systematics when
reconstructing the past history of black-hole binaries using the spin
orientations.
