  The first measurements of the coherence factor R_{K_S^0K\pi} and the average
strong--phase difference \delta^{K_S^0K\pi} in D^0 \to K_S^0 K^\mp\pi^\pm
decays are reported. These parameters can be used to improve the determination
of the unitary triangle angle \gamma\ in B^- \rightarrow $\widetilde{D}K^-$
decays, where $\widetilde{D}$ is either a D^0 or a D^0-bar meson decaying to
the same final state, and also in studies of charm mixing. The measurements of
the coherence factor and strong-phase difference are made using
quantum-correlated, fully-reconstructed D^0D^0-bar pairs produced in e^+e^-
collisions at the \psi(3770) resonance. The measured values are R_{K_S^0K\pi} =
0.70 \pm 0.08 and \delta^{K_S^0K\pi} = (0.1 \pm 15.7)$^\circ$ for an
unrestricted kinematic region and R_{K*K} = 0.94 \pm 0.12 and \delta^{K*K} =
(-16.6 \pm 18.4)$^\circ$ for a region where the combined K_S^0 \pi^\pm
invariant mass is within 100 MeV/c^2 of the K^{*}(892)^\pm mass. These results
indicate a significant level of coherence in the decay. In addition, isobar
models are presented for the two decays, which show the dominance of the
K^*(892)^\pm resonance. The branching ratio {B}(D^0 \rightarrow
K_S^0K^+\pi^-)/{B}(D^0 \rightarrow K_S^0K^-\pi^+) is determined to be 0.592 \pm
0.044 (stat.) \pm 0.018 (syst.), which is more precise than previous
measurements.
