We find two Lax representations for the reduced magnetohydrodynamics equations ({\sc rmhd}) and construct a local variational Poisson structure (a Hamiltonian operator) for them. Its inverse defines a nonlocal symplectic structure for the same equations. We describe the action of both operators on the second-order cosymmetries and on the infinitesimal contact symmetries of {\sc rmhd}, respectively. The reduction of {\sc rmhd} by the symmetry of shifts along the $z$-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics ({\sc imhd}). Applied to the Lax representations and the variational Poisson structure of {\sc rmhd}, the reduction provides analogous constructions for {\sc imhd}.