The first measurements of the forward-backward asymmetry of the dimuon pair (AFB), the triple-product asymmetry (A2ϕ), and the charge-parity-conjugation asymmetry (ACP), in D0→π+π−μ+μ− and D0→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 AFB(D0→π+π−μ+μ−)=(3.3±3.7±0.6)%, A2ϕ(D0→π+π−μ+μ−)=(−0.6±3.7±0.6)%, ACP(D0→π+π−μ+μ−)=(4.9±3.8±0.7)%, AFB(D0→K+K−μ+μ−)=(0±11±2)%, A2ϕ(D0→K+K−μ+μ−)=(9±11±1)%, ACP(D0→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.