  We perform detailed muon spin rotation ($\mu$SR) measurements in the classic
antiferromagnet Fe$_2$O$_3$ and explain the spectra by considering dynamic
population and dissociation of charge-neutral muon-polaron complexes. We show
that charge-neutral muon states in Fe$_2$O$_3$, despite lacking the signatures
typical of charge-neutral muonium centers in nonmagnetic materials, have a
significant impact on the measured $\mu$SR frequencies and relaxation rates.
Our identification of such polaronic muon centers in Fe$_2$O$_3$ suggests that
isolated hydrogen (H) impurities form analogous complexes, and that H
interstitials may be a source of charge carrier density in Fe$_2$O$_3$.
