The first measurements of the forward-backward asymmetry of the dimuon pair ( A FB ), the triple-product asymmetry ( A 2 ϕ ), and the charge-parity-conjugation asymmetry ( A C P ), 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 2 ϕ ( D 0 → π + π − μ + μ − ) = ( − 0.6 ± 3.7 ± 0.6 ) % , A C P ( D 0 → π + π − μ + μ − ) = ( 4.9 ± 3.8 ± 0.7 ) % , A FB ( D 0 → K + K − μ + μ − ) = ( 0 ± 11 ± 2 ) % , A 2 ϕ ( D 0 → K + K − μ + μ − ) = ( 9 ± 11 ± 1 ) % , A C P ( 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.