The first measurements of the forward-backward asymmetry of the dimuon pair (${A}_{\mathrm{FB}}$), the triple-product asymmetry (${A}_{2\ensuremath{\phi}}$), and the charge-parity-conjugation asymmetry (${A}_{CP}$), in ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ decays are reported. They are performed using data from proton-proton collisions collected with the LHCb experiment from 2011 to 2016, corresponding to a total integrated luminosity of $5\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The asymmetries are measured to be ${A}_{\mathrm{FB}}({D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=(3.3\ifmmode\pm\else\textpm\fi{}3.7\ifmmode\pm\else\textpm\fi{}0.6)%$, ${A}_{2\ensuremath{\phi}}({D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=\phantom{\rule{0ex}{0ex}}(\ensuremath{-}0.6\ifmmode\pm\else\textpm\fi{}3.7\ifmmode\pm\else\textpm\fi{}0.6)%$, ${A}_{CP}({D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=(4.9\ifmmode\pm\else\textpm\fi{}3.8\ifmmode\pm\else\textpm\fi{}0.7)%$, ${A}_{\mathrm{FB}}({D}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=\phantom{\rule{0ex}{0ex}}(0\ifmmode\pm\else\textpm\fi{}11\ifmmode\pm\else\textpm\fi{}2)%$, ${A}_{2\ensuremath{\phi}}({D}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=(9\ifmmode\pm\else\textpm\fi{}11\ifmmode\pm\else\textpm\fi{}1)%$, ${A}_{CP}({D}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})=(0\ifmmode\pm\else\textpm\fi{}11\ifmmode\pm\else\textpm\fi{}2)%$, where the first uncertainty is statistical and the second systematic. The asymmetries are also measured as a function of the dimuon invariant mass. The results are consistent with the standard model predictions.