The classical cross-ratio is the unique projective invariant of four points on the line. Consider polygons in $(2N-1)$-dimensional projective symplectic space, satisfying the condition that every $N$ consecutive vertices span a Lagrangian subspace of the parent $2N$-dimensional symplectic space. In the critical case of $(2N+2)$-gons, we will describe a complete set of symplectic invariants of these polygons. We will use only elementary linear algebra; no prior knowledge of symplectic or projective geometry will be assumed.