My gut says that combining that with other sort cycles in some magic ratio could do magical things. I'm quite wrong quite often ofc but the adventure is out there!
Fun fact: Fisher-Yates samples a uniform distribution over permutations and is therefor capable of producing any permutation, so you can sort any array in linear time using Fisher-Yates with its RNG replaced by a appropriate oracle. (So the complexity of sorting a array mostly reduces to the complexity of determining which permutation it's in relative to a sorted version.)