  The first measurements of the forward-backward asymmetry of the dimuon pair
($A_{FB}$), the triple-product asymmetry ($A_{2\phi}$), and the
charge-parity-conjugation asymmetry ($A_{CP}$), in $D^0\to\pi^+\pi^-\mu^+\mu^-$
and $D^0\to K^+K^-\mu^+\mu^-$ 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 \begin{align*}
A_{FB}(D^0\to\pi^+\pi^-\mu^+\mu^-) &= (\phantom{-}3.3\pm3.7\pm0.6)\%,\\
A_{2\phi}(D^0\to\pi^+\pi^-\mu^+\mu^-)&= (-0.6\pm3.7\pm0.6)\%,\\
A_{CP}(D^0\to\pi^+\pi^-\mu^+\mu^-) &= (\phantom{-}4.9\pm3.8\pm0.7)\%,\\
A_{FB}(D^0\to K^+K^-\mu^+\mu^-) &= (0\pm11\pm2)\%,\\ A_{2\phi}(D^0\to
K^+K^-\mu^+\mu^-)&= (9\pm11\pm1)\%,\\ A_{CP}(D^0\to K^+K^-\mu^+\mu^-) &=
(0\pm11\pm2)\%, \end{align*} 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.
