Connections between the structures of Group 12 dihalides in their vapor and crystal phases are sought and discussed. The molecular structures of all monomers and dimers (MX(2): M=Zn, Cd, Hg and X=F, Cl, Br, I) were calculated at the density functional B3PW91 and MP2 computational levels. All the monomers are linear, with the mercury dihalide molecules having shorter bonds than their cadmium analogues; the ZnX(2) and CdX(2) structures are similar. The shorter Hg-X distances are traced back to relativistic effects. For the dimers, many possible geometrical arrangements were considered. The zinc and cadmium dihalide dimers have the usual D(2h)-symmetry geometry, whereas the mercury dihalide dimers are loosely-bound units with C(2h) symmetry. The origins of this C(2h) structure are discussed from different points of view, including frontier orbital interactions. The crystals of Group 12 dihalides span a wide range of structure types, from three-dimensional extended solids to molecular crystals. There is an obvious connection between the structures and characteristics of monomers, their dimers, and the crystals they form. The similarities as well as startling differences from the Group 2 dihalides are analyzed.