CIZA J2242.8+5301, a merging galaxy cluster at z=0.19, hosts a double-relic system and a faint radio halo. Radio observations at frequencies ranging from a few MHz to several GHz have shown that the radio spectral index at the outer edge of the N relic corresponds to a shock of Mach number 4.6+/-1.1, under the assumptions of diffusive shock acceleration of thermal particles in the test particle regime. Here, we present results from new Chandra observations of the cluster. The Chandra surface brightness profile across the N relic only hints to a surface brightness discontinuity (<2-sigma detection). Nevertheless, our reanalysis of archival Suzaku data indicates a temperature discontinuity across the relic that is consistent with a Mach number of 2.5+/-0.5, in agreement with previously published results. This confirms that the Mach number at the shock traced by the N relic is much weaker than predicted from the radio. Puzzlingly, in the Chandra data we also identify additional inner small density discontinuities both on and off the merger axis. Temperature measurements on both sides of the discontinuities do not allow us to undoubtedly determine their nature, although a shock front interpretation seems more likely. We speculate that if the inner density discontinuities are indeed shock fronts, then they are the consequence of violent relaxation of the dark matter cores of the clusters involved in the merger.