The tetrahedron
The tetrahedron has rotational symmetries of order $2$ and $3$.
The poles of order two are the midpoints of edges. The poles
of order $3$ are the vertices, or the centres of the faces (which
are opposite the vertices). The rotation group acts on the four
vertices as the alternating group $A_4$. The full symmetric
group $S_4$ also acts on the tetrahedron, but the odd
permutations act by orthogonal matrices of determinant $-1$,
which are not rotations. The transpositions act as reflections,
but the $4$-cycles are more complicated.
The cube and octahedron
The cube and the octahedron have the same rotational symmetries,
which have order $2$, $3$ or $4$. The poles of order $2$ are the
midpoints of the edges of the cube, which are the same (up to
rescaling) as the midpoints of the edges of the octahedron.
The poles of order $3$ are the vertices of the cube, or the
centres of the faces of the octahedron. The poles of order $4$
are the vertices of the octahedron, or the centres of the faces
of the cube. The rotation group acts on the four long diagonals
of the cube as the permutation group $S_4$.
The dodecahedron and icosahedron
The dodecahedron and the icosahedron have the same rotational
symmetries, which have order $2$, $3$ or $5$. The poles of order
$2$ are the midpoints of the edges of the dodecahedron, which are
the same (up to rescaling) as the midpoints of the edges of the
icosahedron. The poles of order $3$ are the vertices of the
dodecahedron, or the centres of the faces of the icosahedron..
The poles of order $5$ are the vertices of the icosahedron, or
the centres of the faces of the dodecahedron. The rotation group
acts on the five inscribed cubes of the dodecahedron as the
permutation group $S_4$.