The relation between magnetic activity and rotation in late-type stars provides fundamental information on stellar dynamos and angular momentum evolution. Rotation-activity studies found in the literature suffer from inhomogeneity in the measurement of activity indexes and rotation periods. We overcome this limitation with a study of the X-ray emitting, late-type main-sequence stars observed by XMM-Newton and Kepler. We measured rotation periods from photometric variability in Kepler light curves. As activity indicators, we adopted the X-ray luminosity, the number frequency of white-light flares, the amplitude of the rotational photometric modulation, and the standard deviation in the Kepler light curves. The search for X-ray flares in the light curves provided by the EXTraS (Exploring the X-ray Transient and variable Sky) FP-7 project allows us to identify simultaneous X-ray and white-light flares. A careful selection of the X-ray sources in the Kepler field yields 102 main-sequence stars with spectral types from A to M. We find rotation periods for 74 X-ray emitting main-sequence stars, 20 of which do not have period reported in the previous literature. In the X-ray activity-rotation relation, we see evidence for the traditional distinction of a saturated and a correlated part, the latter presenting a continuous decrease in activity towards slower rotators. For the optical activity indicators the transition is abrupt and located at a period of ~10 d but it can be probed only marginally with this sample, which is biased towards fast rotators due to the X-ray selection. We observe seven bona-fide X-ray flares with evidence for a white-light counterpart in simultaneous Kepler data. We derive an X-ray flare frequency of ~0.15 d⁻¹, consistent with the optical flare frequency obtained from the much longer Kepler time-series.