  Inverse muon decay ($\nu_\mu e^- \to \nu_e \mu^-$) is a promising tool to
constrain neutrino fluxes with energies $E_{\nu} \ge 10.9~\mathrm{GeV}$.
Radiative corrections introduce percent-level distortions to energy spectra of
outgoing muons and depend on experimental details. In this paper, we calculate
radiative corrections to the scattering processes $\nu_\mu e^- \to \nu_e \mu^-$
and $\bar{\nu}_e e^- \to \bar{\nu}_\mu \mu^-$. We present the muon energy
spectrum for both channels, double-differential distributions in muon energy
and muon scattering angle and in photon energy and photon scattering angle, and
the photon energy spectrum for the dominant $\nu_\mu e^- \to \nu_e \mu^-$
process. Our results clarify and extend the region of applicability of previous
results in the literature for the double differential distribution in muon
energy and photon energy, and in the muon energy spectrum with a radiated
photon above a threshold energy. We provide analytic expressions for single,
double and triple differential cross sections, and discuss how radiative
corrections modify experimentally interesting observable distributions.
