Rotationally symmetric plabic graphs and the Lagrangian Grassmannian

We introduce the totally nonnegative Lagrangian Grassmannian ${\mathrm{L}\mathrm{G}}_{\ge 0}^R(n,2n)$, a new subset of the totally nonnegative Grassmannian consisting of subspaces isotropic with respect to a certain bilinear form $R$. We describe its cell structure and show that each cell admits a representation by a rotationally symmetric (not necessarily reduced) plabic graph. Along the way, we develop new techniques for working with non-reduced plabic graphs.