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.