Not sure whether this is what you are after, but one way to give a geometric meaning is the following.

You can think of $P^1{\mathbb C}$ as the "boundary at infinity" of the 3-dimensional hyperbolic space. To every "ideal tetrahedron" (i.e., a tetrahedron in hyperbolic 3-space with vertices at infinity) you can associate the cross ratio of its four vertices.

(The best way to think of the cross ration is probably that the cross ratio of the 4 points $0,1,z,\infty$ is $z$ and that the cross ratio is invariant under the action of $PSL(2,{\mathbb C})$. Since every non-degenerate 4-tuple is of the form $A0, A1, Az, A\infty$ for some $A\in PSL(2,{\mathbb C})$, this determines the cross ratio uniquely.)

So to every non-degenerate tetrahedron with vertices at infinity you get a cross ratio in $P^1{\mathbb C}$. (Actually for a non-degenerate terahedron you only get elements in $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$.)

Now, the symmetric group $S_4$ acts by permutation on the four vertices of a tetrahedron. You get an induced action on the possible cross ratios (i.e., on $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$) and you can check that there are only three possibilities, either $z$ is mapped to itself, or to $$\frac{1}{1-z}$$ or to $$1-\frac{1}{z}.$$(The action extends in the obvious way to $\left\{0,1,\infty\right\}$.) Let $A\subset S_4$ be the stabilizer of some $z$, then $A\backslash S_4$ is isomorphic to ${\mathbb Z}/3{\mathbb Z}$.

So the natural action of $S_4$ by permuting vertices of tetrahedra factors over an action of ${\mathbb Z}/3{\mathbb Z}$ which is exactly the action you described above.

Not sure whether this is what you are after, but one way to give a geometric meaning is the following.

You can think of $P^1{\mathbb C}$ as the "boundary at infinity" of the 3-dimensional hyperbolic space. To every "ideal tetrahedron" (i.e., a tetrahedron in hyperbolic 3-space with vertices at infinity) you can associate the cross ratio of its four vertices.

(The best way to think of the cross ration is probably that the cross ratio of the 4 points $0,1,z,\infty$ is $z$ and that the cross ratio is invariant under the action of $PSL(2,{\mathbb C})$. Since every non-degenerate 4-tuple is of the form $A0, A1, Az, A\infty$ for some $A\in PSL(2,{\mathbb C})$, this determines the cross ratio uniquely.)

So to every non-degenerate tetrahedron with vertices at infinity you get a cross ratio in $P^1{\mathbb C}$. (Actually for a non-degenerate terahedron you only get elements in $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$.)

Now, the symmetric group $S_4$ acts by permutation on the four vertices of a tetrahedron. You get an induced action on the possible cross ratios (i.e., on $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$) and you can check that for even permutations $\sigma\in A_4$ there are only three possibilities, either $z$ is mapped to itself, or to $$\frac{1}{1-z}$$ or to $$1-\frac{1}{z}.$$(The action extends in the obvious way to $\left\{0,1,\infty\right\}$.) Let $Stab(z)\subset A_4$ be the stabilizer of some $z$, then $Stab(z)\backslash A_4$ is isomorphic to ${\mathbb Z}/3{\mathbb Z}$.

So the natural action of $A_4\subset S_4$ by permuting vertices of tetrahedra factors over an action of ${\mathbb Z}/3{\mathbb Z}$ which is exactly the action you described above.

You'll find more detail in https://en.wikipedia.org/wiki/Cross-ratio#Six_cross-ratios_and_the_anharmonic_group

Not sure whether this is what you are after, but one way to give a geometric meaning is the following.

You can think of $P^1{\mathbb C}$ as the "boundary at infinity" of the 3-dimensional hyperbolic space. To every "ideal tetrahedron" (i.e., a tetrahedron in hyperbolic 3-space with vertices at infinity) you can associate the cross ratio of its four vertices.

(The best way to think of the cross ration is probably that the cross ratio of the 4 points $0,1,z,\infty$ is $z$ and that the cross ratio is invariant under the action of $PSL(2,{\mathbb C})$. Since every non-degenerate 4-tuple is of the form $A0, A1, Az, A\infty$ for some $A\in PSL(2,{\mathbb C})$, this determines the cross ratio uniquely.)

So to every non-degenerate tetrahedron with vertices at infinity you get a cross ratio in $P^1{\mathbb C})$. (Actually for a non-degenerate terahedron you only get elements in $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$.

Now, the symmetric group $S_4$ acts by permutation on the four vertices of a tetrahedron. You get an induced action on the possible cross ratios (i.e., on $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$) and you can check that there are only three possibilities, either $z$ is mapped to itself, or to $$\frac{1}{1-z}$$ or to $$1-\frac{1}{z}.$$(The action extends in the obvious way to $\left\{0,1,\infty\right\}$.) Let $A\subset S_4$ be the stabilizer of some $z$, then $A\backslash S_4$ is isomorphic to ${\mathbb Z}/3{\mathbb Z}$.

So the natural action of $S_4$ by permuting vertices of tetrahedra factors over an action of ${\mathbb Z}/3{\mathbb Z}$ which is exactly the action you described above.

Not sure whether this is what you are after, but one way to give a geometric meaning is the following.

You can think of $P^1{\mathbb C}$ as the "boundary at infinity" of the 3-dimensional hyperbolic space. To every "ideal tetrahedron" (i.e., a tetrahedron in hyperbolic 3-space with vertices at infinity) you can associate the cross ratio of its four vertices.

(The best way to think of the cross ration is probably that the cross ratio of the 4 points $0,1,z,\infty$ is $z$ and that the cross ratio is invariant under the action of $PSL(2,{\mathbb C})$. Since every non-degenerate 4-tuple is of the form $A0, A1, Az, A\infty$ for some $A\in PSL(2,{\mathbb C})$, this determines the cross ratio uniquely.)

So to every non-degenerate tetrahedron with vertices at infinity you get a cross ratio in $P^1{\mathbb C}$. (Actually for a non-degenerate terahedron you only get elements in $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$.)

Now, the symmetric group $S_4$ acts by permutation on the four vertices of a tetrahedron. You get an induced action on the possible cross ratios (i.e., on $P^1{\mathbb C}\setminus\left\{0,1,\infty\right\}$) and you can check that there are only three possibilities, either $z$ is mapped to itself, or to $$\frac{1}{1-z}$$ or to $$1-\frac{1}{z}.$$(The action extends in the obvious way to $\left\{0,1,\infty\right\}$.) Let $A\subset S_4$ be the stabilizer of some $z$, then $A\backslash S_4$ is isomorphic to ${\mathbb Z}/3{\mathbb Z}$.

So the natural action of $S_4$ by permuting vertices of tetrahedra factors over an action of ${\mathbb Z}/3{\mathbb Z}$ which is exactly the action you described above.