  A plabic graph is a planar bicolored graph embedded in a disk, which
satisfies some combinatorial conditions. Postnikov's boundary measurement map
takes the space of positive edge weights of a plabic graph $G$ to a positroid
cell in some totally nonnegative Grassmannian. In this note, we investigate
plabic graphs which are symmetric about a line of reflection, up to reversing
the colors of vertices. These symmetric plabic graphs arise naturally in the
study of total positivity for the Lagrangian Grassmannian. We characterize
various combinatorial objects associated with symmetric plabic graphs, and
describe the subset of a Grassmannian which can be realized by symmetric
weightings of symmetric plabic graphs.
