Several of the $M{\mathrm{Cd}}_6$ approximants $(M=\mathrm{Pr},$ Nd, Sm, Eu, Gd, Dy, Yb, Y, and Ca) to the $M{\mathrm{Cd}}_{5.7}$ quasicrystals $(M=\mathrm{Yb}$ and Ca) have been synthesized and their structures have been refined from single-crystal x-ray-diffraction data; the different types of disorder of the central ${\mathrm{Cd}}_4$ tetrahedra located in the dodecahedral cavities of these closely related compounds have been examined and mutually compared. A model that describes this disorder and can be applied to all the examined $M{\mathrm{Cd}}_6$ phases has been elaborated. The refinements made on these phases differ from what is found in previous reports.