A study is made of the superconformal transformation properties of the recently constructed $O(2,3)$-invariant classical solutions of the massless Wess–Zumino model. It is shown that these properties are completely determined by two supersubgroups $O\operatorname{Sp}(1,4)$ of the superconformal group which intersect on the subgroup $O(2,3)$. One $O\operatorname{Sp}(1,4)$ is the stability subgroup of the solutions. The other $O\operatorname{Sp}(1,4)$ is spontaneously broken down to $O(2,3)$. Its odd transformations uniquely fix the dependence of the solutions on the Grassmann degrees of freedom and generate the complete set of solutions. We note a possible connection between the $O\operatorname{Sp}(1,4)$ structure of the Wess–Zumino model and the analogous structure in spontaneously broken supergravity, and we discuss ways of generalizing our results to theories with Euclidean supersymmetry.