  The doped perovskite BaBiO$_3$ exhibits a maximum superconducting transition
temperature ($T_c$) of 34 K and was the first high-$T_c$ oxide to be
discovered, yet pivotal questions regarding the nature of both the metallic and
superconducting states remain unresolved. Although it is generally thought that
superconductivity in the bismuthates is of the conventional s-wave type, the
pairing mechanism is still debated, with strong electron-phonon coupling and
bismuth valence or bond disproportionation possibly playing a role. Here we use
diffuse x-ray scattering and Monte Carlo modeling to study the local structure
of Ba$_{1-x}$K$_x$BiO$_3$ across its insulator-metal boundary. We find no
evidence for either long- or short-range disproportionation, which resolves a
major conundrum, as disproportionation and the related polaronic effects are
likely not relevant for the metallic and superconducting states. Instead, we
uncover nanoscale structural correlations that break inversion symmetry, which
has far-reaching implications for the electronic physics, including the pairing
mechanism. This unexpected finding furthermore establishes that the bismuthates
belong to the broader classes of materials with hidden spin-orbit coupling and
a tendency towards inversion-breaking displacements.
