  Within the framework of 2PN black-hole binary spin precession, we explore
configurations where one of the two spins oscillates from being completely
aligned with the orbital angular momentum to being completely anti-aligned with
it during a single precession cycle. This "wide nutation" is the extreme limit
of the generic phenomenon of spin nutation in black-hole binaries. Crucially,
wide nutation happens on the short precession time scale and it is not a
secular effect due to gravitational-wave radiation reaction. The spins of these
binaries, therefore, flip repeatedly as one of these special configurations is
entered. Binaries with total mass $M$, mass ratio $q$, and dimensionless spin
$\chi_1$ ($\chi_2$) of the more (less) massive black hole are allowed to
undergo wide nutation at binary separations $r \leq r_{\rm wide} \equiv [(q
\chi_2 - \chi_1)/(1-q)]^2 M$. Sources that are more likely to nutate widely
have similar masses and effective spins close to zero.
