  We use a combination of neutron and X-ray total scattering measurements
together with pair distribution function (PDF) analysis to characterise the
variation in local structure across the orbital order--disorder transition in
LaMnO$_3$. Our experimental data are inconsistent with a conventional
order--disorder description of the transition, and reflect instead the
existence of a discontinuous change in local structure between ordered and
disordered states. Within the orbital-ordered regime, the neutron and X-ray
PDFs are best described by a local structure model with the same local orbital
arrangements as those observed in the average (long-range) crystal structure.
We show that a variety of meaningfully-different local orbital arrangement
models can give fits of comparable quality to the experimental PDFs collected
within the disordered regime; nevertheless, our data show a subtle but
consistent preference for the anisotropic Potts model proposed in \emph{Phys
Rev.\ B} {\bf 79}, 174106 (2009). The key implications of this model are
electronic and magnetic isotropy together with the loss of local inversion
symmetry at the Mn site. We conclude with a critical assessment of the
interpretation of PDF measurements when characterising local symmetry breaking
in functional materials.
