Abstract: "From the hundreds of gamma-ray pulsars known, only a handful show non-thermal X-ray pulsations. Instead, nine objects pulse in non-thermal X-rays but lack counterparts at higher energies. Here, we present a physical model for the non-thermal emission of pulsars above 1 keV. With just four physical parameters, we fit the spectrum of the gamma/X-ray pulsars along seven orders of magnitude. We find that all detections can be encompassed in a continuous variation of the model parameters, and pose that their values could likely relate to the closure mechanism operating in the accelerating region. The model explains the appearance of sub-exponential cutoffs at high energies as a natural consequence of synchro-curvature dominated losses, unveiling that curvature-only emission may play a relatively minor role --if any-- in the spectrum of most pulsars. The model also explains the flattening of the X-ray spectra at soft energies as a result of propagating particles being subject to synchrotron losses all along their trajectories. Using this model, we show how observations in gamma-rays can predict the detectability of the pulsar in X-rays, and viceversa."