Dipositronium, Ps2, was recently prepared . This is significant because: • It is the first laboratory observation of a molecule that contains more than one positron; • It is the most symmetrical molecule possible; • It is the most non-rigid (floppiest) molecule possible; • The interval between the theoretical establishment of its existence  and its laboratory observation  is inordinately long – 60 years; and • An extension of the technology developed for the Ps2 observation may soon lead to an observation of the Bose-Einstein condensation of positronium and the development of a gamma ray laser. We briefly discuss the symmetry of Ps2 and how an understanding of it will underlie its characterization in the future. Ps2O and CPs2 might be the next two-positron compounds to be prepared and characterized in the laboratory. A discussion of the contrasting eigenstates of these two molecules is given. An understanding of these states is required in order to identify them.