Let's say that we have finite field $\mathbb F_q$ and we have a couple of involutions $g,f$ with exactly one fixed point (zero).

Let's take any element $\alpha \in \mathbb F_q$ Let's start applying involutions to this element in the following way: $$ \ldots f(g(\alpha)) \ldots g(\alpha) \ldots \alpha \ldots f(\alpha) \ldots g(f(\alpha)) \ldots $$ Let's imagine that on some step the element on the left side become equal to the element on the right side for any $\alpha$.

What is the name for such couple of involutions with this property ?

Are there any theorems regarding existence of such couple of involutions ?

The example of such couple of involutions is here: Chains of numbers generated by 2 involutions.