What can one learn about material structure given a single first-principles calculation?

We extract a variable X from electron orbitals Psi_nk and energies E_nk in the parent high-symmetry structure of a wide range of complex oxides (perovskites, rutiles, pyrochlores, cristobalites). Even though X was calculated in the parent structure we show that its value dictates material's (potentially lower-symmetry) ground state structure. We propose using Wannier functions to extract concealed variables such as X both for material structure prediction and for high-throughput approaches.