Substitutional defects in a cubic symmetry (such as a lithium defect in a KCl host crystal) can be modelled appropriately by an eight-state system. Usually this tunnelling degree of freedom is approximated by a two-level system. We investigate the observable differences between the two models in three contexts. First, we show that the two models predict different relations between the temperature dependence of the specific heat and that of the static susceptibility. Second, we demonstrate that in the presence of external forces (pressure and electric field) the eight-state system shows features that cannot be understood within the framework of the two-level approximation. In this context we propose an experiment for measuring the parameter governing tunnelling along the face diagonal. Finally, we discuss the differences between the models appearing for strongly coupled pairs. Geometric selection rules and particular forms of asymmetry lead to clear differences between the two models.