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In this paper, we introduce the notions of state-space symmetry for nonlinear systems and of the cross operator as a nontrivial natural extension of the linear symmetric case, in terms of the controllability and observability operators associated to it. We give a characterization of a symmetric nonlinear system in terms of the cross operator and a coordinate transformation. Then we analyze the use of the cross operator for solving the Hankel singular value problem of the system. The result is a new and simpler characterization of the solutions of this problem in terms of the cross operator and a metric.