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 $r\le rwide\equiv [(q\chi 2-\chi 1)/(1-q)]2M$. Sources that are more likely to nutate widely have similar masses and effective spins close to zero.