The idea of the coordination polyhedron is essential to understanding chemical structure. Simple polyhedra in crystalline compounds are often deformed due to structural complexity or electronic instabilities so distortion analysis methods are useful. Here we demonstrate that analysis of the minimum bounding ellipsoid of a coordination polyhedron provides a general method for studying distortion, yielding parameters that are sensitive to various orders in metal oxide examples. Ellipsoidal analysis leads to discovery of a general switching of polyhedral distortions at symmetry-disallowed transitions in perovskites that may evidence underlying coordination bistability, and reveals a weak off-centre ‘d5 effect’ for Fe3+ ions that could be exploited in multiferroics. Separating electronic distortions from intrinsic deformations within the low temperature superstructure of magnetite provides new insights into the charge and trimeron orders. Ellipsoidal analysis can be useful for exploring local structure in many materials such as coordination complexes and frameworks, organometallics and organic molecules.