Abstract Using K2, we recently discovered a new type of periodic photometric variability while analyzing the light curves of members of Upper Sco. The 23 exemplars of this new variability type are all mid-M dwarfs, with short rotation periods. Their phased light curves have one or more broad flux dips or multiple arcuate structures which are not explicable by photospheric spots or eclipses by solid bodies. Now, using Transiting Exoplanet Survey Satellite data, we have searched for this type of variability in the other major sections of Sco-Cen, Upper Centaurus-Lupus (UCL), and Lower Centaurus-Crux (LCC). We identify 28 stars with the same light curve morphologies. We find no obvious difference between the Upper Sco and the UCL/LCC representatives of this class in terms of their light curve morphologies, periods, or variability amplitudes. The physical mechanism behind this variability is unknown, but as a possible clue we show that the rapidly rotating mid-M dwarfs in UCL/LCC have slightly different colors from the slowly rotating M dwarfs—they either have a blue excess (hot spots?) or a red excess (warm dust?). One of the newly identified stars (TIC242407571) has a very striking light curve morphology. At about every 0.05 in phase are features that resemble icicles. The icicles arise because there is a second periodic system whose main feature is a broad flux dip. Using a toy model, we show that the observed light curve morphology results only if the ratio of the two periods and the flux-dip width are carefully arranged.