We designed nine new cage compounds by using different numbers of linkages -N(NO2)- and -N(NH2)- to connect the two skeletons of 1,3,5-trinitro-1,3,5-triazinane (RDX) or 1,3,5,7-tetranitro-1,3,5,7-tetrazocane (HMX) to construct the cage frameworks, similar to the skeleton of hexanitrohexaazaisowurtzitane (CL-20). Then, their electronic structures, heats of formation, energetic properties, and impact sensitivities were studied using density functional theory. The types of intermolecular interactions and the changes in sensitivity were analyzed. Compared to the parent compounds RDX, HMX, and CL-20, all the designed compounds present extraordinary high detonation properties (D > 9.70 km/s and P > 44.30 GPa). Considering the detonation performance and impact sensitivity, the four compounds may be regarded as potential candidates of high-energy-density compounds . Our results will contribute to a new strategy for the theoretical design of new energetic compounds.