  This paper is concerned with so-called index $d$ generalized cyclotomic
mappings of a finite field $\mathbb{F}_q$, which are functions
$\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial
function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of
$\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in the
context of generalized cyclotomic mappings and present applications thereof
that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$
and pertain to cycle structures, the classification of $(q-1)$-cycles and
involutions, as well as inversion.
