  We perform a symmetry analysis of the ABC-stacked kagome planes of Mn atoms
common to the $L1_2$ Mn$_3$X and antiperovskite Mn$_3$AB alloys. In addition to
the single-ion-anisotropy and Kitaev-type anisotropic exchange known to
stabilize 120{\deg} spin structures in these materials, our analysis results in
a staggered Dzyaloshinskii-Moriya interaction and further gamma-type
anisotropic exchange between nearest-neighbor spins. The presence of these new
terms is shown not to affect the energetics of the 120{\deg} ground-states
which explains their absence in prior minimal magnetic models. We go on to
show, however, that their influence becomes apparent when spin-wave excitations
are considered. We highlight these effects by calculating inelastic neutron
scattering cross-sections to illustrate experimental means of probing the
existence and relative strengths of these cloaked interactions.
