We analyze the inspiral dynamics of equal-mass precessing black-hole binaries using multi-timescale techniques. The orbit-averaged post-Newtonian evolutionary equations admit two constants of motion in the equal-mass limit, namely the magnitude of the total spin S and the effective spin ξ. This feature makes the entire dynamics qualitatively different compared to the generic unequal-mass case, where only ξ is constant while the variable S parametrizes the precession dynamics. For fixed individual masses and spin magnitudes, an equal-mass black-hole inspiral is uniquely characterized by the two parameters : these two numbers completely determine the entire evolution under the effect of radiation reaction. In particular, for equal-mass binaries we find that (i) the black-hole binary spin morphology is constant throughout the inspiral, and that (ii) the precessional motion of the two black-hole spins about the total spin takes place on a longer timescale than the precession of the total spin and the orbital plane about the total angular momentum.