The general problem of the structure of the low-lying motional states of impurity species in crystals is approached by applying a group-theoretical reduction scheme to a basis set consisting of nearly isolated "pocket" states in a rigid lattice. The reduction is carried out for a number of possible symmetries. The concept of "barrier-lowering" perturbation is used within the framework of first-order perturbation theory to treat the ordering and splitting of motional states. The method is applied to Li+, OH-, and CN- as impurity species in KCl, with particular emphasis on the analysis of dielectric measurements. It is concluded, in particular, that motion of OH- involves both rotational and translational modes.