We report on a multi-wavelength study of the compact object candidate 1RXS J141256.0+792204 (Calvera). Calvera was observed in the X-rays with XMM/EPIC twice for a total exposure time of ~50 ks. The source spectrum is thermal and well reproduced by a two component model composed of either two hydrogen atmosphere models, or two blackbodies (kT_1~ 55/150 eV, kT_2~ 80/250 eV, respectively, as measured at infinity). Evidence was found for an absorption feature at ~0.65 keV; no power-law high-energy tail is statistically required. Using pn and MOS data we discovered pulsations in the X-ray emission at a period P=59.2 ms. The detection is highly significant (> 11 sigma), and unambiguously confirms the neutron star nature of Calvera. The pulse profile is nearly sinusoidal, with a pulsed fraction of ~18%. We looked for the timing signature of Calvera in the Fermi Large Area Telescope (LAT) database and found a significant (~5 sigma) pulsed signal at a period coincident with the X-ray value. The gamma-ray timing analysis yielded a tight upper limit on the period derivative, dP/dt < 5E-18 s/s (dE_rot/dt <1E33 erg/s, B<5E10 G for magneto- dipolar spin-down). Radio searches at 1.36 GHz with the 100-m Effelsberg radio telescope yielded negative results, with a deep upper limit on the pulsed flux of 0.05 mJy. Diffuse, soft (< 1 keV) X-ray emission about 13 west of the Calvera position is present both in our pointed observations and in archive ROSAT all-sky survey images, but is unlikely associated with the X-ray pulsar. Its spectrum is compatible with an old supernova remnant (SNR); no evidence for diffuse emission in the radio and optical bands was found. The most likely interpretations are that Calvera is either a central compact object escaped from a SNR or a mildly recycled pulsar; in both cases the source would be the first ever member of the class detected at gamma-ray energies.
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