Crazy Cosine Carousel

We define a carousel to be a (unordered) set of four pairwise distinct real numbers $\{t_1,t_2,t_3,t_4\},$ all strictly between $0$ and $2\pi ,$ such that in some order they satisfy the following system of equations

$\begin{cases} \cos (2t_1) = 4\cos t_1 \cos t_2\\ \cos (2t_2) = 4\cos t_2 \cos t_3\\ \cos (2t_3) = 4\cos t_3 \cos t_4\\ \cos (2t_4) = 4\cos t_4 \cos t_1. \end{cases}$

How many carousels are there?

Details and assumptions

A set is unordered. If a set satisfies the above system for several different orderings, it is only counted once.

A set of values is called pairwise distinct if no two of them are equal. For example, the set $\{1, 2, 2\}$ is not pairwise distinct, because the last two values are the same.