A submanifold \(M\) of an Euclidean space is \(2\)-symmetric if for each \(x\in M\) there exists an involutory isometry \(s_x\) of the ambient space, with fixed point \(x\) isolated on \(M\) and mapping locally \(M\) onto itself. All totally geodesic submanifolds of a symmetric submanifold of an Euclidean space are \(2\)-symmetric. The notion was proposed by \textit{O. Kowalski} and \textit{I. Kulich} [Math. Ann. 277, 67-88 (1987; Zbl 0614.53040)], then studied mainly in [\textit{A. Carfagna, R. Mazzocco} and \textit{G. Romani}, Czech. Math. J. 44, No. 4, 691-711 (1994; Zbl 0823.53022); \textit{A. Carfagna} and \textit{G. Romani}, Ann. Mat. Pura Appl., IV. Ser. 162, 237-252 (1992; Zbl 0786.53033)] in relation with the ``nicely curved'' property of submanifolds. In the same spirit, here the authors characterize the totally geodesic maps from a nicely curved submanifold \(M\) of \({\mathbb R}^n\) into a Grassmannian by means of the \(2\)-symmetry of \(M\). The last section of the paper is devoted to an explicit construction of examples of minimal symmetric generalized ruled submanifolds and \(2\)-symmetric submanifolds.