Quantum-correlated $psi(3770) to Dbar{D}$ decays collected by the CLEO-c experiment are used to perform a first measurement of $F_+^{4pi}$, the fractional $CP$-even content of the self-conjugate decay $D to pi^+pi^-pi^+pi^-$, obtaining a value of $0.737 pm 0.028$. An important input to the measurement comes from the use of $D to K^0_{rm S}pi^+pi^-$ and $D to K^0_{rm L}pi^+pi^-$ decays to tag the signal mode. This same technique is applied to the channels $D topi^+pi^-pi^0$ and $D to K^+K^-pi^0$, yielding $F_+^{pipipi^0} = 1.014 pm 0.045 pm 0.022$ and $F_+^{KKpi^0} = 0.734 pm 0.106 pm 0.054$, where the first uncertainty is statistical and the second systematic. These measurements are consistent with those of an earlier analysis, based on $CP$-eigenstate tags, and can be combined to give values of $F_+^{pipipi^0} = 0.973 pm 0.017$ and $F_+^{KKpi^0} = 0.732 pm 0.055$. The results will enable the three modes to be included in a model-independent manner in measurements of the unitarity triangle angle $gamma$ using $B^mp to DK^mp$ decays, and in time-dependent studies of $CP$ violation and mixing in the $Dbar{D}$ system.
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