Nonlinear commutativity preserving maps on self-adjoint operators

We characterize the bijective nonlinear maps of the set of all self-adjoint bounded linear operators on a complex separable Hilbert space H of dimension at least 3 which preserve commutativity in both directions. Roughly speaking, a bijective map has this property if, and only if, up to a unitary or antiunitary transformation of H, it leaves fixed the self-adjoint parts of the commutative von Neumann algebras on H.