Radial velocity monitoring has found the signature of a Msini = 1.3M⊙ planet located within the habitable zone (HZ) of Proxima Centauri. Despite a hotter past and an active host star, the planet Proxima b could have retained enough volatiles to sustain surface habitability. Here we use a 3D Global Climate Model (GCM) to simulate the atmosphere and water cycle of Proxima b for its two likely rotation modes (1:1 and 3:2 spin-orbit resonances), while varying the unconstrained surface water inventory and atmospheric greenhouse effect. Any low-obliquity, low-eccentricity planet within the HZ of its star should be in one of the climate regimes discussed here. We find that a broad range of atmospheric compositions allow surface liquid water. On a tidally locked planet with sufficient surface water inventory, liquid water is always present, at least in the substellar region. With a non-synchronous rotation, this requires a minimum greenhouse warming (~10 mbar of CO2 and 1 bar of N2). If the planet is dryer, ~0.5 bar or 1.5 bars of CO2 (for asynchronous or synchronous rotation, respectively) suffice to prevent the trapping of any arbitrary, small water inventory into polar or nightside ice caps. We produce reflection and emission spectra and phase curves for the simulated climates. We find that atmospheric characterization will be possible via direct imaging with forthcoming large telescopes. The angular separation of 7λ/D at 1 μm (with the E-ELT) and a contrast of ~10-7 will enable high-resolution spectroscopy and the search for molecular signatures, including H2O, O2, and CO2. The observation of thermal phase curves can be attempted with the James Webb Space Telescope, thanks to a contrast of 2 × 10-5 at 10 μm. Proxima b will also be an exceptional target for future IR interferometers. Within a decade it will be possible to image Proxima b and possibly determine whether the surface of this exoplanet is habitable.