  We derive several existence results concerning cycle types and, more
generally, the "mapping behavior" of complete mappings. Our focus is on
so-called first-order cyclotomic mappings, which are functions on a finite
field $\mathbb{F}_q$ that fix $0$ and restrict to the multiplication $x\mapsto
a_ix$ by a fixed element $a_i\in\mathbb{F}_q$ on each coset $C_i$ of a given
subgroup $C$ of $\mathbb{F}_q^{\ast}$. The gist of two of our main results is
that as long as $q$ is large enough relative to the index
$|\mathbb{F}_q^{\ast}:C|$, all cycle types of first-order cyclotomic
permutations with only long cycles on $\mathbb{F}_q^{\ast}$ can be achieved
through a complete mapping, as can all permutations of the cosets of $C$. Our
third main result provides new examples of complete mappings $f$ such that both
$f$ and its associated orthomorphism $f+\operatorname{id}$ permute the nonzero
field elements in one cycle.
