We analyze the behavior of the mixed Hellan-Herrmann-Johnson method for solving the biharmonic problem Δ 2 ψ = f {\Delta ^2}\psi = f . We show a superconvergence result for the distance between ψ h {\psi ^h} (the approximation of the displacement) and P h ψ {P_h}\psi (where P h {P_h} is a suitable projection operator). If the discrete equations are solved (as is usually done) by using interelement Lagrange multipliers, our superconvergence result allows us to prove the convergence, in suitable norms, of the Lagrange multipliers to the normal derivative of the displacement, and to construct a new approximation of ∇ ψ \nabla \psi which converges to ∇ ψ \nabla \psi faster than ∇ ψ h \nabla {\psi ^h} .