A new article titled Crystal Structure and Electronic Properties of Bulk and Thin Film Brownmillerite Oxides by Josh Young and Prof. Rondinelli outlines the key structural features that determine polymorphic stability in oxides with corner-connected tetrahedra and octahedra.
The work which appears in Phys. Rev. B 92, 174111 (2015) also describes how epitaxial strain can be used to tip the balance among the competing interactions captured in the schematic shown to the right.