{gamma}-ray radiation from pulsars is usually thought to be mostly produced by the synchro-curvature (SC) losses of accelerated particles. Here, we present a systematic study of all currently reported, good-quality Fermi-LAT pulsar spectral data. We do so by applying a model which follows the particle dynamics and consistently computes the emission of SC radiation. By fitting observational data on a case by case basis, we are able to obtain constraints about the parallel electric field, the typical length-scale over which particles emit the bulk of the detected radiation, and the number of involved particles. The model copes well with data of several dozens of millisecond (MSPs) and young pulsars (YPs). By correlating the inferred model parameters with the observed timing properties, some trends are discovered. First, a non-negligible part of the radiation comes from the loss of perpendicular momentum soon after pair creation. Second, the electric field strongly correlates with both the inverse of the emission length-scale and the magnetic field at light cylinder, thus ruling out models with high-energy photon production close to the surface. These correlations unify young and millisecond pulsars under the same physical scenario, and predict that magnetars are intrinsically {gamma}-ray quiet via synchro-curvature processes, since magnetospheric particles are not accelerated enough to emit a detectable {gamma}-ray flux.