  We develop new and precise geometric descriptions of the conjugacy class
$[x]$ and coconjugation set $\operatorname{C}(x,x') = \{ y \in \overline{W}
\mid yxy^{-1} = x' \}$ for all elements $x,x'$ of any affine Coxeter group
$\overline{W}$. The centralizer of $x$ in $\overline{W}$ is the special case
$\operatorname{C}(x,x)$. The key structure in our description of the conjugacy
class $[x]$ is the mod-set ${Mod}_{\overline{W}}(w) =
(w-\operatorname{I})R^\vee$, where~$w$ is the finite part of $x$ and $R^\vee$
is the coroot lattice. The coconjugation set $\operatorname{C}(x,x')$ is then
described by ${Mod}_{\overline{W}}(w')$ together with the fix-set of $w'$,
where $w'$ is the finite part of $x'$. For any element $w$ of the associated
finite Weyl group $W$, the mod-set of $w$ is contained in the classical
move-set ${Mov}(w) = \operatorname{Im}(w - \operatorname{I})$. We prove that
the rank of ${Mod}_{\overline{W}}(w)$ equals the dimension of ${Mov}(w)$, and
then further investigate type-by-type the surprisingly subtle structure of the
$\mathbb{Z}$-module ${Mod}_\overline{W}(w)$. As corollaries, we determine
exactly when ${Mod}_{\overline{W}}(w) = {Mov}(w) \cap R^\vee$, in which case
our closed-form descriptions of conjugacy classes and coconjugation sets are as
simple as possible.
