  We measure the branching fraction for the Singly Cabibbo-suppressed decay
$D^0\rightarrow K^0_S\,K^0_S\,\pi^+\pi^-$, and we search for $CP$ violation via
a measurement of the $CP$ asymmetry $A^{}_{CP}$ and also the $T$-odd
triple-product asymmetry $a_{CP}^{T}$. The later two measurements are
complementary. We use 922~fb$^{-1}$ of data recorded by the Belle experiment,
which ran at the KEKB asymmetric-energy $e^+ e^-$ collider. The branching
fraction is measured relative to the Cabibbo-favored normalization channel
$D^0\rightarrow K^0_S\,\pi^+\pi^-$. Singly Cabibbo-suppressed charm decays are
expected to have especially good sensitivity to new physics effects.
