We have analyzed K2 light curves for more than 3000 low-mass stars in the ~8 Myr old Upper Sco association, the ~125 Myr age Pleiades open cluster, and the ~700 Myr old Hyades and Praesepe open clusters to determine stellar rotation rates. Many of these K2 targets show two distinct periods, and for the lowest-mass stars in these clusters, virtually all of these systems with two periods are photometric binaries. The most likely explanation is that we are detecting the rotation periods for both components of these binaries. We explore the evolution of the rotation rate in both components of photometric binaries relative to one another and to nonphotometric binary stars. In Upper Sco and the Pleiades, these low-mass binary stars have periods that are much shorter on average and much closer to each other than would be true if drawn at random from the M dwarf single stars. In Upper Sco, this difference correlates strongly with the presence or absence of infrared excesses due to primordial circumstellar disks-the single-star population includes many stars with disks, and their rotation periods are distinctively longer on average than their binary star cousins of the same mass. By Praesepe age, the significance of the difference in rotation rate between the single and binary low-mass M dwarf stars is much less, suggesting that angular momentum loss from winds for fully convective zero-age main-sequence stars erases memory of the rotation rate dichotomy for binary and single very low mass stars at later ages.