While two-dimensional triangles are always subdivision invariant, the same does not always hold for their three-dimensional counterparts. We consider several interesting properties of those three-dimensional tetrahedra which are subdivision invariant and offer them a new classification. Moreover, we study the optimization of these tetrahedra, arguing that the second Sommerville tetrahedra are the closest to being regular and are optimal by many measures. Anisotropic subdivision invariant tetrahedra with high aspect ratios are characterized. Potential implications and applications of our findings are also discussed.