  We report the first measurement of the $T$-odd moments in the decay $D^{0}
\rightarrow K_{S}^{0} \pi^{+} \pi^{-} \pi^{0}$ from a data sample corresponding
to an integrated luminosity of $966\,{\rm fb}^{-1}$ collected by the Belle
experiment at the KEKB asymmetric-energy $e^+ e^-$ collider. From these moments
we determine the $CP$-violation-sensitive asymmetry $a_{CP}^{T\text{-odd}} =
\left[-0.28 \pm 1.38 ~(\rm{stat.}) ^{+0.23}_{-0.76} ~(\rm{syst.})\right] \times
10^{-3}$, which is consistent with no $CP$ violation. In addition, we perform
$a_{CP}^{T\text{-odd}}$ measurements in different regions of the $D^{0}
\rightarrow K_{S}^{0} \pi^{+} \pi^{-} \pi^{0}$ phase space; these are also
consistent with no $CP$ violation.
