The first measurements of the forward-backward asymmetry of the dimuon pair (A_{FB}), the triple-product asymmetry (A_{2ϕ}), and the charge-parity-conjugation asymmetry (A_{CP}), in D^{0}→π^{+}π^{-}μ^{+}μ^{-} and D^{0}→K^{+}K^{-}μ^{+}μ^{-} 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  fb^{-1}. The asymmetries are measured to be A_{FB}(D^{0}→π^{+}π^{-}μ^{+}μ^{-})=(3.3±3.7±0.6)%, A_{2ϕ}(D^{0}→π^{+}π^{-}μ^{+}μ^{-})=(-0.6±3.7±0.6)%, A_{CP}(D^{0}→π^{+}π^{-}μ^{+}μ^{-})=(4.9±3.8±0.7)%, A_{FB}(D^{0}→K^{+}K^{-}μ^{+}μ^{-})=(0±11±2)%, A_{2ϕ}(D^{0}→K^{+}K^{-}μ^{+}μ^{-})=(9±11±1)%, A_{CP}(D^{0}→K^{+}K^{-}μ^{+}μ^{-})=(0±11±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.