Many physical systems can be mapped onto solved or "solvable" models of magnetism. In this work, we have mapped the statistical mechanics of columnar phases of ideally helical rigid DNA -subject to the earlier found unusual, frustrated pair potential (A.A. Kornyshev, S. Leikin, J. Chem. Phys. 107, 3656 (1997))- onto an exotic, unknown variant of the XY model on a fixed or restructurable lattice. Here, the role of the "spin" is played by the azimuthal orientation of the molecules. We have solved this model using a Hartree-Fock approximation, ground-state calculations, and finite-temperature Monte Carlo simulations. We have found peculiar spin order transitions, which may also be accompanied by positional restructuring, from hexagonal to rhombohedric lattices. Some of these have been experimentally observed in dense columnar aggregates. Note that DNA columnar phases are of great interest in biophysical research, not only because they are a useful in vitro tool for the study of DNA condensation, but also since these structures have been detected in living matter. Within the approximations made, our study provides insight into the statistical mechanics of these systems.