A neutral dark matter (DM) particle with a magnetic dipole moment has a very different direct detection phenomenology with respect to standard candidates. This is due to the peculiar functional form of the differential cross section for scattering with nuclei. Such a candidate could be a bound state of charged particles, as the neutron or an atom, or a fundamental particle coupled to heavier charged states, much like a Dirac neutrino. We analyze here the direct detection signals of DM with magnetic dipole moment, both the recoil rate and its modulation, and show that they are very different from those expected in standard scenarios. For this candidate, contrary to the common lore, the time of maximum signal depends on the recoil energy as well as on the target material. The observation of different modulations by experiments employing different targets would be a strong indication in favor of this type of DM particles.