This is an exercise from the book "Visual Group Theory" by Nathan Carter.
Exercise $5.26$. As you know from the chapter, the symmetry group for the tetrahedron is $A_4$. We can think of it, as you saw in Exercise $5.25$, as permuting four vertices. What physical features of the tetrahedron prevent its symmetry group from being all of $S_4$?
Intuitively, $(A\ D)$ is not in the symmetry group for the tetrahedron. However, how to justify this more rigorously (e.g., mathematically maybe in terms of geometry)? For example, can we justify this in terms of isometry? (I am not sure.)