The present article studies the duality of the Lagrange problem of optimal control theory with the boundary value constraints given by second-order polyhedral differential inclusions. Our primary aim is to establish results of duality for a boundary value problem with second-order differential inclusions. As a supplementary problem, we consider differential problems and formulate sufficient conditions of optimality, including particular transversality conditions incorporating the Euler-Lagrange type inclusions. After constructing the dual problem for second-order polyhedral differential inclusions, we prove that the adjoint Euler-Lagrange inclusion is simultaneously a dual relationship, which is satisfied by the pair of solutions of the primal and dual problems. Furthermore, solving numerical examples illustrates the application of these results.