We report asymmetries in the forward-angle differential cross sections for pion single-charge-exchange scattering from oriented and unoriented $^{165}\mathrm{Ho}$ nuclei. These asymmetries are directly related to the neutron deformation of a nucleus that has a large charge quadrupole moment. The measured asymmetry extrapolated to 0\ifmmode^\circ\else\textdegree\fi{} is -0.020\ifmmode\pm\else\textpm\fi{}0.024. In the model of Chiang and Johnson, this value implies a quadrupole deformation ratio ${\mathrm{\ensuremath{\beta}}}_{2}^{\mathit{n}}$/${\mathrm{\ensuremath{\beta}}}_{2}^{\mathit{p}}$=0.84\ifmmode\pm\else\textpm\fi{}0.08, indicating that the excess neutrons are less deformed than the protons. This result conflicts with the predictions of the best available Hartree-Fock models based on Skyrme interactions. We also report on asymmetries of the nonresonant continuum and of the giant dipole resonances.