Three new ligands containing a bicyclo[2.2.2]oct‐7‐ene‐2,3,5,6‐tetracarboxydiimide unit have been used to assemble lantern‐type metal‐organic cages with the general formula [Cu4L4]. Functionalisation of the backbone of the ligands leads to distinct crystal packing motifs between the three cages, as observed with single‐crystal X‐ray diffraction. The three cages vary in their gas sorption behaviour, and the capacity of the materials for CO2 is found to depend on the activation conditions: softer activation conditions lead to superior uptake, and one of the cages displays the highest BET surface area found for lantern‐type cages so far.