Calvera is an isolated neutron star detected only through its thermal X-ray emission. Its location at high Galactic latitude (b = +37°) is unusual if Calvera is a relatively young pulsar, as suggested by its timing parameters. Using the Neutron Star Interior Composition Explorer, we obtained a phase-connected timing solution spanning four years, which allowed us to measure the second and the third derivative of the frequency, and to reveal timing noise consistent with that of normal radio pulsars. A magnetized hydrogen atmosphere model, covering the entire star surface, provides a good description of the phase-resolved spectra and energy-dependent pulsed fraction, but a temperature distribution steeper than isotropic is needed. The inferred distance is ~3.3 kpc, that places Calvera well above the Galactic disk. Our results favor the hypothesis that Calvera is a normal rotation-powered pulsar, that was born in the Galactic halo.