This paper is concerned with state-space realizations of the adjoints of the variationals of Hamiltonian control systems. It will be shown that the variational systems of a class of Hamiltonian systems have self-adjoint state-space realizations, that is, the variational system and its adjoint have the same state-space realizations. This implies that the input-output mapping of the adjoint of the variational system of a given Hamiltonian system can be calculated by only using the input-output mapping of the original system. Furthermore, this property is applied to adjoint based iterative learning control with optimal control type cost functions.