Let $\square ABCD$ be a simple quadrilateral. Now there are many ways to represent the same quadrilateral, such as $\square BCDA$ or $\square ADCB$. These notations can be described by permutations; for example, the following permutation will correspond to $\square BCDA$. $$\sigma=\begin{pmatrix}A & B & C & D \\ B & C & D & A\end{pmatrix}$$ However, $\square ABDC$ is not equal to $\square ABCD$, since the sides are not equal.
What I want to do is to find all the possible notations for $\square ABCD$. I've worked my way in proving that $\square ADCB$ is indeed equal to $\square ABCD$, and I want to prove that "pushing" the vertices leads to the same quadrilateral. In other words, for $P_1, P_2, P_3, P_4\in\{A,B,C,D\}$, $\square \sigma(P_1)\sigma(P_2)\sigma(P_3)\sigma(P_4)$ is equal to $\square P_1P_2P_3P_4$. Any help will be appreciated.
Basically you have all circular permutations of $ABCD$
$\begin{array}{} A&B&C&D\\ &B&C&D&A\\ &&C&D&A&B\\ &&&D&A&B&C\\ \end{array}$
Similarly when the polygon is walked in reverse orientation, circular permutations of $DCBA$.
$\begin{array}{} D&C&B&A\\ &C&B&A&D\\ &&B&A&D&C\\ &&&A&D&C&B\\ \end{array}$
So in cycle notations these are $(ABCD)$ and $(DCBA)$.