Knutson and Zinn-Justin gave a geometric explanation of the Knutson-Tao puzzle rule for computing Schubert calculus of certain partial flag varieties in type A . The geometry is based on Lagrangian correspondences between Nakajima quiver varieties. We extend Knutson and Zinn-Justin's construction to triple products of Nakajima quiver varieties. By defining a compatible Z/2Z action on our quiver varieties, we furthermore construct Lagrangian correspondences between cotangent bundles of Grassmannians and symplectic Grassmannians. This is the first, and hardest, step towards a puzzle rule for computing the product of symplectic Grassmannian Schubert classes by Grassmannian Schubert classes.