We consider the elementary operator $M_{A,B}$, acting on the Hilbert-Schmidt class $C_{2}\left( {\mathcal{H}}\right) $, given by $M_{A,B}\left( T\right) =ATB$ with $A$ and $B$ bounded operators on ${\mathcal{H}}$. In this work, we establish necessary and sufficient conditions on $A$ and $B$ for $M_{A,B}$ to be a $2$-symmetric and $3$-symmetric. We also characterize binormality of elementary operators.