  We measure the spin-density matrix elements (SDMEs) of the
$\Delta^{++}(1232)$ in the photoproduction reaction $\gamma p \to
\pi^-\Delta^{++}(1232)$ with the GlueX experiment in Hall D at Jefferson Lab.
The measurement uses a linearly--polarized photon beam with energies from $8.2$
to $8.8$~GeV and the statistical precision of the SDMEs exceeds the previous
measurement by three orders of magnitude for the momentum transfer squared
region below $1.4$ GeV$^2$. The data are sensitive to the previously
undetermined relative sign between couplings in existing Regge-exchange models.
Linear combinations of the extracted SDMEs allow for a decomposition into
natural and unnatural--exchange amplitudes. We find that the unnatural exchange
plays an important role in the low momentum transfer region.
