Gemini surfactants comprise two single-tailed surfactants connected by a linker at or near the hydrophilic headgroup. They display a variety of water-concentration-dependent lyotropic liquid crystal morphologies that are sensitive to surfactant molecular structure and the nature of the headgroups and counterions. Recently, an interesting dependence of the aqueous-phase behavior on the length of the linker has been discovered; odd-numbered linker length surfactants exhibit characteristically different phase diagrams than even-numbered linker surfactants. In this work, we investigate this "odd/even effect" using computer simulations, focusing on experimentally studied gemini dicarboxylates with Na+ counterions, seven nonterminal carbon atoms in the tails, and either three, four, five, or six carbon atoms in the linker (denoted Na-73, Na-74, Na-75, and Na-76, respectively). We find that the relative electrostatic repulsion between headgroups in the different morphologies is correlated with the qualitative features of the experimental phase diagrams, predicting destabilization of hexagonal phases as the cylinders pack close together at low water content. Significant differences in the relative headgroup orientations of Na-74 and Na-76 compared to those of Na-73 and Na-75 surfactants lead to differences in linker-linker packing and long-range headgroup-headgroup electrostatic repulsion, which affects the delicate electrostatic balance between the hexagonal and gyroid phases. Much of the fundamental insight presented in this work is enabled by the ability to computationally construct and analyze metastable phases that are not observable in experiments.