I have seen statements of the form: "such-and-such is characterized by a 4-tuple, which is unique up to cyclic permutations." The intended meaning is that we don't want to count the 4-tuples $(x,y,z,w)$ and $(w,x,y,z)$ as really different for our purposes, whereas $(w,z,x,y)$ is different. The idea is that each 4-tuple could be represented up to 4 different ways (or fewer, in cases such as $(x,y,x,y)$ or $(x,x,x,x)$).
Someone pointed out to me, though, the phrase doesn't really say what we want it to say. The permutation that changes $(w,x,y,z)$ to $(w,z,x,y)$ is, in fact, a cyclic permutation - of three of the elements. Indeed, the permutation that turns $(w,x,y,z)$ into $(y,w,z,x)$ is a 4-cycle, that just happens to go in a different order than the order of the 4-tuple.
Is there a more precise way of saying what is meant here than "unique up to cyclic permutation", that isn't also horribly clunky? Or is there a good argument for why "unique up to cyclic permutation" really does mean what I want it to mean?
I hope this question makes sense.