We measure the branching fraction for the Cabibbo-suppressed decay ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and search for $CP$ violation via a measurement of the $CP$ asymmetry ${A}_{CP}$ as well as the $T$-odd triple-product asymmetry ${a}_{CP}^{T}$. We use $922\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data recorded by the Belle experiment, which ran at the KEKB asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider. The branching fraction is measured relative to the Cabibbo-favored normalization channel ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$; the result is $\mathcal{B}({D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=[4.79\ifmmode\pm\else\textpm\fi{}0.08(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.10(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.31(\mathrm{norm})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, where the first uncertainty is statistical, the second is systematic, and the third is from uncertainty in the normalization channel. We also measure ${A}_{CP}=[\ensuremath{-}2.51\ifmmode\pm\else\textpm\fi{}1.44(\mathrm{stat}{)}_{\ensuremath{-}0.10}^{+0.11}(\mathrm{syst})]%$, and ${a}_{CP}^{T}=[\ensuremath{-}1.95\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0ex}{0ex}}1.42(\mathrm{stat}{)}_{\ensuremath{-}0.12}^{+0.14}(\mathrm{syst})]%$. These results show no evidence of $CP$ violation.
