  A measurement of the $C\!P$-even fraction of the decay
$D^{0}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ is performed with a quantum-correlated
$\psi(3770)\to D\bar{D}$ data sample collected by the BESIII experiment,
corresponding to an integrated luminosity of 2.93 $\textrm{fb}^{-1}$. Using a
combination of $C\!P$ eigenstates, $D \to \pi^+\pi^-\pi^0$ and $D \to
K_{S,L}^0\pi^{+}\pi^{-}$ as tagging modes, the $C\!P$-even fraction is measured
to be $F_{+}^{4\pi} = 0.735 \pm 0.015 \pm 0.005$, where the first uncertainty
is statistical and the second is systematic. This is the most precise
determination of this quantity to date. It provides valuable model-independent
input for the measurement of the CKM angle $\gamma$ with $B^\pm\to D K^\pm$
decays, and for time-dependent studies of $C\!P$ violation and mixing in the
$D^{0}$-$\bar{D}^{0}$ system.
