  Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams
defined a set $\mathsf{NC}(a,b)$ of `rational noncrossing partitions', which
form a subset of the ordinary noncrossing partitions of $\{1, 2, \dots, b-1\}$.
Confirming a conjecture of Armstrong et. al., we prove that $\mathsf{NC}(a,b)$
is closed under rotation and prove an instance of the cyclic sieving phenomenon
for this rotational action. We also define a rational generalization of the
$\mathfrak{S}_a$-noncrossing parking functions of Armstrong, Reiner, and
Rhoades.
