Consider the macroscale modelling of fine scale spatiotemporal dynamics. A new approach ensures coarse scale discrete models preserve important self-adjoint properties of the microscale dynamics. Dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids). Such mapping of discrete lattice dynamics may be iterated to empower us in future research to explore scale dependent emergent phenomena. The support of dynamical systems, centre manifold theory ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain and for all time. The accuracy of the models is limited by the asymptotic resolution of subgrid coarse-scale processes, and is controlled by the level of truncation. As given examples demonstrate, the approach developed here ensures the preservation of important conservation properties of the fine scale dynamics.