Results on the production of antihyperons in the reactions of 3.7 BeV/c antiprotons with protons are presented. Examples were observed of the production of the antiparticles of all the baryons with hypercharge 0 and 1, several of which had not been observed prior to this experiment. The mass and lifetime of the $\overline{\ensuremath{\Lambda}}$ and the lifetimes of the ${\overline{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ were measured and are consistent with the corresponding quantities of the $\ensuremath{\Lambda}$ and ${\ensuremath{\Sigma}}^{\ifmmode\pm\else\textpm\fi{}}$, as required by $\mathrm{CPT}$ invariance. In the two-body $\ensuremath{\Lambda}\overline{\ensuremath{\Lambda}}$, $\ensuremath{\Lambda}{\overline{\ensuremath{\Sigma}}}^{0}$, $\overline{\ensuremath{\Lambda}}{\ensuremath{\Sigma}}^{0}$, and ${\ensuremath{\Sigma}}^{+}{\overline{\ensuremath{\Sigma}}}^{\ensuremath{-}}$ final states the antihyperons displayed a very sharply forward peaked angular distribution. The production of the three-body final states $Y\overline{Y}\ensuremath{\pi}$ and $\ensuremath{\Lambda}\overline{N}K$ or $\overline{\ensuremath{\Lambda}}N\overline{K}$ tend to proceed through the $Y{\overline{Y}}^{*}$ or $\overline{Y}{Y}^{*}$ intermediate states, where the ${Y}^{*}$ is either the ${{Y}_{1}}^{*}(1385)$, the ${{Y}_{0}}^{*}(1405)$, or the ${{Y}_{0}}^{*}(1520)$. The four-body $\ensuremath{\Lambda}\overline{\ensuremath{\Lambda}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ final state is dominated by the pair production of the ${{Y}_{1}}^{*}(1385)$ hyperon isobar.