The geometric structure, symmetry, and spin of ${\mathrm{Au}}_{13}$ clusters are investigated in the framework of density-functional theory, with particular attention paid to the correlation among these properties. Several computational approaches are carefully tested on previously proposed cluster configurations. Complications and possible pitfalls in electronic-structure calculations on these systems are highlighted. Using molecular dynamics with quantum mechanically calculated forces, a set of favored (high binding energy) geometric structures, where a trigonal prism acts as the central building block, is discussed.