  We study binary spinning black holes to display the long term individual spin
dynamics. We perform a full numerical simulation starting at an initial proper
separation of $d\approx25M$ between equal mass holes and evolve them down to
merger for nearly 48 orbits, 3 precession cycles, and half of a flip-flop
cycle. The simulation lasts for $t=20000M$ and displays a total change in the
orientation of the spin of one of the black holes from initially aligned with
the orbital angular momentum to a complete anti-alignment after half of a
flip-flop cycle. We compare this evolution with an integration of the 3.5
Post-Newtonian equations of motion and spin evolution to show that this process
continuously flip-flops the spin during the lifetime of the binary until
merger. We also provide lower order analytic expressions for the maximum
flip-flop angle and frequency. We discuss the effects this dynamics may have on
spin growth in accreting binaries and on the observational consequences for
galactic and supermassive binary black holes.
