A solvothermal reaction of CuI with a bifunctional organic ligand 3,5-pyridinedicarboxylic acid (H₂pdc) afforded a novel heterovalent copper–organic framework, [(Cu^I₂I₂)Cu^II₄(pdc)₄(H₂O)₄]·4DMSO (1) (DMSO = dimethyl sulfoxide). Compound 1 was structurally characterized by single-crystal X-ray diffraction analyses and further characterized by infrared spectroscopy (IR), powder X-ray diffraction (PXRD), elemental analyses and thermogravimetric analyses (TGA). Structural analysis indicates that two kinds of common secondary building units (SBUs) coexist in this 3D network. The linkage of Cu⁺ and I⁻ generates [Cu^I₂I₂] subunits and the connection of Cu²⁺ with carboxyl-O atoms gives rise to binuclear paddle-wheel [Cu^II₂(COO)₄(H₂O)₂] moieties. In compound 1, two different SBUs are bridged with pdc²⁻ ligands to construct a micro-sized cage with an internal spherical cavity. Through a cage-to-cage connection, the 3D porous framework is formed which features intersecting 1D channels. Gas sorption measurements were conducted on the activated 1, showing a H₂ uptake of 1.46 wt% at 77 K and under 1 bar with a high initial adsorption enthalpy of 8.2 kJ mol⁻¹. Moreover, compound 1 has excellent adsorption selectivity for CO₂ over N₂ and CH₄.