Studies on several geometrical configurations of the free doubly negative carbon cluster C-9(2-) are performed at the Hartree-Fock level as well as at the configuration interaction and outer-valence Green’s function levels of theory. Most of the C-9(2-) isomers inspected are found to be stable with respect to fragmentation, but only two structures, a branched cluster and a D-3h symmetrical isomer, emerge also as stable with respect to electron autoejection. The characteristic properties which allow small free dianionic carbon clusters to bind two excess electrons are discussed. Qualitative structural criteria for the existence of these cluster dianions are extracted by comparing and generalizing the results for C-7(2-), C-8(2-), C-9(2-), and C-10(2-). Predictions of possible structures of stable higher carbon cluster dianions are made.