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We present the results of an extensive study of the phase diagram and spin wave excitations for a general spin model on a hexagonal AB-stacked kagome system. The boundaries of the magnetic phases are determined via a combination of numerical (Monte Carlo) and analytical (Luttinger-Tisza) methods. We also determine the phase coexistence regions by considering the instabilities in the spin wave spectra. Depending on the strength of the spin-orbit coupling (SOC), some spin and lattice rotations become decoupled, leading to considerably larger symmetry groups than typical magnetic groups. Thus, we provide a detailed symmetry description of the magnetic Hamiltonian with negligible, weak, and intermediate strength of SOC. The spin symmetry in these three cases has a strong effect on the splittings observed in the spin excitation spectra. We further identify a number of self-duality transformations that map the Hamiltonian onto itself. These transformations describe the symmetry of the parameter space and provide an exact mapping between the properties of different magnetic orders and lead to accidental degeneracies. Finally, we discuss the physical relevance of our findings in the context of $\mathrm{Mn}_3X$ compounds.