  During the first few hundred days after the explosion, core-collapse
supernovae (SNe) emit down-scattered X-rays and gamma-rays originating from
radioactive line emissions, primarily from the $^{56}$Ni $\rightarrow$
$^{56}$Co $\rightarrow$ $^{56}$Fe chain. We use SN models based on
three-dimensional neutrino-driven explosion simulations of single stars and
mergers to compute this emission and compare the predictions with observations
of SN 1987A. A number of models are clearly excluded, showing that high-energy
emission is a powerful way of discriminating between models. The best models
are almost consistent with the observations, but differences that cannot be
matched by a suitable choice of viewing angle are evident. Therefore, our
self-consistent models suggest that neutrino-driven explosions are able to
produce, in principle, sufficient mixing, although remaining discrepancies may
require small changes to the progenitor structures. The soft X-ray cutoff is
primarily determined by the metallicity of the progenitor envelope. The main
effect of asymmetries is to vary the flux level by a factor of ${\sim}$3. For
the more asymmetric models, the shapes of the light curves also change. In
addition to the models of SN 1987A, we investigate two models of Type II-P SNe
and one model of a stripped-envelope Type IIb SN. The Type II-P models have
similar observables as the models of SN 1987A, but the stripped-envelope SN
model is significantly more luminous and evolves faster. Finally, we make
simple predictions for future observations of nearby SNe.
