We give some characterizations of dual Toeplitz operators acting on the orthogonal complement of the Bergman space over bounded symmetric domains. Our main result characterizes those finite sums of products of Toeplitz operators that are themselves dual Toeplitz operators. Furthermore, we obtain a necessary condition for such finite sums of dual Toeplitz products to be compact. As an application of our main result, we derive a sufficient and necessary condition for when the (semi-)commutators of dual Toeplitz operators is zero. Notably, we find that a dual Toeplitz operator is compact if and only if it is the zero operator.