Shape-memory effects are quite well-studied in general, but there is only one reported example in the context of porous materials. We report the second example of a porous coordination network that exhibits a sorbate-induced shape-memory effect and the first in which multiple sorbates, N2, CO2 and CO promote this effect. The material, a new threefold interpenetrated pcu network, [Zn2(4,4'-biphenyldicarboxylate)2(1,4-bis(4-pyridyl)benzene)]n (X-pcu-3-Zn-3i), exhibits three distinct phases: the as-synthesized α phase; a denser-activated β phase; and a shape-memory γ phase, which is intermediate in density between the α and β phases. The γ phase is kinetically stable over multiple adsorption/desorption cycles and only reverts to the β phase when heated at >400 K under vacuum. The α phase can be regenerated by soaking the γ phase in N,N'-dimethylformamide. Single-crystal x-ray crystallography studies of all three phases provide insight into the shape-memory phenomenon by revealing the nature of interactions between interpenetrated networks. The β and γ phases were further investigated by in situ coincidence powder x-ray diffraction, and their sorption isotherms were replicated by density functional theory calculations. Analysis of the structural information concerning the three phases of X-pcu-3-Zn-3i enabled us to understand structure-function relationships and propose crystal engineering principles for the design of more examples of shape-memory porous materials.