We study the relation between stellar rotation and magnetic activity for a sample of 134 bright, nearby M dwarfs observed in the Kepler Two-Wheel (K2) mission during campaigns C0 to C4. The K2 lightcurves yield photometrically derived rotation periods for 97 stars (79 of which without previous period measurement), as well as various measures for activity related to cool spots and flares. We find a clear difference between fast and slow rotators with a dividing line at a period of ~10d at which the activity level changes abruptly. All photometric diagnostics of activity (spot cycle amplitude, flare peak amplitude and residual variability after subtraction of spot and flare variations) display the same dichotomy, pointing to a quick transition between a high-activity mode for fast rotators and a low-activity mode for slow rotators. This unexplained behavior is reminiscent of a dynamo mode-change seen in numerical simulations that separates a dipolar from a multipolar regime. A substantial number of the fast rotators are visual binaries. A tentative explanation is accelerated disk evolution in binaries leading to higher initial rotation rates on the main-sequence and associated longer spin-down and activity lifetimes. We combine the K2 rotation periods with archival X-ray and UV data. X-ray, FUV and NUV detections are found for 26, 41, and 11 stars from our sample, respectively. Separating the fast from the slow rotators, we determine for the first time the X-ray saturation level separately for early- and for mid-M stars.