  We provide a cyclic permutation analogue of the Erd\H os-Szekeres theorem. In
particular, we show that every cyclic permutation of length $(k-1)(\ell-1)+2$
has either an increasing cyclic sub-permutation of length $k+1$ or a decreasing
cyclic sub-permutation of length $\ell+1$, and show that the result is tight.
We also characterize all maximum-length cyclic permutations that do not have an
increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic
sub-permutation of length $\ell+1$.
