  A search for time-integrated $CP$ violation in the Cabibbo-suppressed decay
$\mbox{$D^0\rightarrow\pi^+\pi^-\pi^+\pi^-$}$ is performed using an unbinned,
model-independent technique known as the energy test. This is the first
application of the energy test in four-body decays. The search is performed for
$P$-even $CP$ asymmetries and, for the first time, is extended to probe the
$P$-odd case. Using proton-proton collision data corresponding to an integrated
luminosity of 3.0 fb$^{-1}$ collected by the LHCb detector at centre-of-mass
energies of $\sqrt{s}=$7 TeV and 8 TeV, the world's best sensitivity to $CP$
violation in this decay is obtained. The data are found to be consistent with
the hypothesis of $CP$ symmetry with a $p$-value of $(4.6\pm0.5)\%$ in the
$P$-even case, and marginally consistent with a $p$-value of $(0.6\pm0.2)\%$ in
the $P$-odd case, corresponding to a significance for $CP$ non-conservation of
2.7 standard deviations.
