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

We extract a variable $X$ from electron orbitals $\Psi_{n\bf{k}}$ and energies $E_{n\bf{k}}$ in the parent high-symmetry structure of a wide range of complex oxides: perovskites, rutiles, pyrochlores, and cristobalites. Even though calculation was done only in the parent structure, with no distortions, we show that $X$ dictates material's true 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.