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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 56Ni → 56Co → 56Fe chain. We use supernova (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 ∼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 SNe II-P and one model of a stripped-envelope SN IIb. The Type II-P models have observables similar to those of 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.