In this paper, we propose a novel design and fabrication strategy to produce architected structures for use as the core in composite sandwich structures. A traditional foam core or honeycomb structure is lightweight and stiff, but susceptible to permanent deformation when subjected to excessive loading. Here we propose the use of an architected structure composed of arrays of hollow truncated cone unit cells that dissipate energy and exhibit structural recovery. These structures, printed with a viscoelastic material, rely on buckling of their sidewalls to dissipate energy and snap-back to prevent permanent deformation. We explore the mechanical response of these conical unit cells in terms of their buckling strength and post-buckling stability condition and develop design maps for the same by relating them to non-dimensional geometric parameters α,β,γ where α represents the slenderness of the curved sidewalls, β is the angle of the sidewall to the base, and γ represents the curvature of the sidewall. A validated finite element model is developed and used to investigate the effect of these parameters. We show that the volume normalized peak buckling load is directly proportional to both α & β, but is independent of γ. The post-buckling stability is influenced by γ as a large radius of curvature makes the structure less likely to exhibit structural bistability. Due to viscoelastic dissipation, for certain geometric parameters, the structures exhibit pseudo-bistability which allows them to recover to their original configurations without the need for external stimuli or energy.