  Calvera (1RXS J141256.0+792204) is an isolated neutron star detected only
through its thermal X-ray emission. Its location at high Galactic latitude
($b=+37^\circ$) is unusual if Calvera is a relatively young pulsar, as
suggested by its spin period (59 ms) and period derivative ($3.2 \times
10^{-15}$ Hz s$^{-1}$). Using the Neutron Star Interior Composition Explorer
(NICER), we obtained a phase-connected timing solution spanning four years
which allowed us to measure the second derivative of the frequency $\ddot\nu =
-2.5 \times 10^{-23}$ Hz s$^{-2}$ 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. However, we find that a
temperature map more anisotropic than that produced by a dipole field is
required, with a hotter zone concentrated towards the poles. By adding two
small polar caps, we find that the surface effective temperature and that of
the caps are ~0.1 and ~0.36 keV, respectively. The inferred distance is ~3.3
kpc. We confirm the presence of an absorption line at 0.7 keV associated to the
emission from the whole star surface, difficult to interpret as a cyclotron
feature and more likely originating from atomic transitions. We searched for
pulsed $\gamma$-ray emission by folding seven years of Fermi-LAT data using the
X-ray ephemeris, but no evidence for pulsations was found. Our results favour
the hypothesis that Calvera is a normal rotation-powered pulsar, with the only
peculiarity of being born at a large height above the Galactic disk.
