I fully agree with @HungryMonkey7 (and ChatGPT, for once 
).
When restricting to rationals a/q with 1 <= a <= q-1, the tricot is a permutation. There are 2 cases:
- If 2 is a primitive root mod q, then there is a unique cycle with gcd(a,q)=1 of length phi(q) where phi(q) is the Euler totient function (number of numerators a coprime to q). This case occurs for q=3, 5, 9, 11, 13, 19, 25, … and phi(q)=q-1 whenever q is prime.
- Otherwise, all cycles with gcd(a,q)=1 are of the same length which is the order of 2 in (Z/qZ)^x. This occurs for q=7, 15, 17, 21, 23,… with respective number of cycles 2, 2, 2, 2, 2, … and lengths 3, 4, 8, 6, 11, … The number of cycles can be larger than 2, e.g., for q=31, there are 6 cycles of lengths 5.
See https://en.wikipedia.org/wiki/Primitive_root_modulo_n for further details.