  This contribution compiles the benefits of lattice symmetry in the context of
closed orbit correction. A symmetric arrangement of BPMs and correctors results
in structured orbit response matrices of Circulant or block Circulant type.
These forms of matrices provide favorable properties in terms of computational
complexity, information compression and interpretation of mathematical vector
spaces of BPMs and correctors. For broken symmetries, a nearest-Circulant
approximation is introduced and the practical advantages of symmetry
exploitation are demonstrated with the help of simulations and experiments in
the context of FAIR synchrotrons.
