  We perform quantum-correlated measurements of the decay $D \to K^{0}_{\rm
S}\pi^{+}\pi^{-}\pi^{0}$ using a data sample corresponding to an integrated
luminosity of 0.82 fb$^{-1}$ collected at the $\psi(3770)$ resonance by the
CLEO-c detector. The value of the $CP$-even fraction $F_{+}$ is determined to
be 0.238 $\pm$ 0.012 $\pm$ 0.012. The strong-phase differences are also
measured in different regions of $K^{0}_{\rm S}\pi^{+}\pi^{-}\pi^{0}$ phase
space by binning around the intermediate resonances present. The potential
sensitivity of the results for determining the CKM angle $\gamma$ from $B^{\pm}
\to D(K^{0}_{\rm S}\pi^{+}\pi^{-}\pi^{0})K^{\pm}$ decays is also discussed.
