  We investigate the structure of the fixed-point algebra of $\mathcal{O}_n$
under the action of the cyclic permutation of the generating isometries. We
prove that it is $*$-isomorphic with $\mathcal{O}_n$, thus generalizing a
result of Choi and Latr\'emoli\`ere on $\mathcal{O}_2$. As an application of
the technique employed, we also describe the fixed-point algebra of
$\mathcal{O}_{2n}$ under the exchange automorphism.
