  $\gamma$-ray radiation from pulsars is usually thought to be mostly produced
by the synchro-curvature 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 synchro-curvature radiation. By
fitting observational data on a case by case basis, we are able to obtain
constraints about the parallel electric field, the typical lengthscale 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 and young pulsars. 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 lengthscale 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 syncrhro-curvature processes, since
magnetospheric particles are not accelerated enough to emit a detectable
$\gamma$-ray flux.
