Origami and kirigami inspired foldcores have recently seen a surge in research and engineering applications mainly due to their open channel configuration to avoid moisture accumulation and diversification of design parameters to meet customized requirements. In this paper, a new kirigami-inspired pyramid foldcore was proposed, and the performance of the corresponding sandwich structure subjected to quasi-static out-of-plane compression and shear, respectively, were studied experimentally, numerically, and analytically. The deformation mechanism of the new sandwich in each loading scenario was revealed, and the effects of geometric parameters were obtained through parametric analysis. Moreover, a theoretical model was established to estimate the average shear stress of the sandwich. In comparison with the commonly adopted square honeycomb, Miura-ori, and eggbox sandwiches, the average compressive stress of the new sandwich was respectively increased by 73%, 342%, and 130%. Upon shear loading, the average shear stress of the new sandwich outperformed the Miura-ori by 34%, and was comparable with that of the square honeycomb by a difference of 11%. To summarize, the new foldcore shows great promise in the design of high-performance sandwich structures.