A continuous symmetry study of the structures of transition metal six-vertex polyhedra is presented, considering both molecular models and experimental structural data. The concept of symmetry map is introduced, consisting of a scatterplot of the symmetry measures relative to two alternative ideal polyhedra. In the case of hexacoordinated complexes, we take as reference shapes the octahedron and the equilateral trigonal prism and study different distortions from these two extremes, including the Bailar twist that interconverts one into another. Such a symmetry map allows us to establish trends in the structural chemistry of the coordination sphere of hexacoordinated transition metal atoms, including the effects of several factors, such as the electron configuration or the presence of bidentate, terdentate or encapsulating ligands. Also introduced is the concept of a symmetry constant, which identifies a distortive route that preserves the minimum distance to two reference symmetries. A wide variety of model distortions are analyzed, and the models are tested against experimental structural data of a wide variety of six-coordinated complexes.