$\DeclareMathOperator\GF{GF}$Consider the finite field $\GF(p)$ for prime $p$.

Consider the pair of involutions $f(x) = 1-x$ , $g(x) = 1/x$, and the chain of numbers generated by these 2 involutions in the following way: $$\cdots f(g(f(x))) \leftarrow g(f(x)) \leftarrow f(x) \to x \to g(x) \to f(g(x) \to g(f(g(x)) \cdots$$

Apparently the maximal length of this chain for specific $x$ is equal to 6.

Could you please explain if this construction has some special name in mathematics, or was studied in the theory of finite fields?

For example for $\GF(31)$ we have:

\begin{gather*} 12 \leftarrow 20 \leftarrow 14 \to 18 \to 19 \to 13 \to 12 \\ 12, 13, 14, 18, 19, 20. \end{gather*}