  We describe the geometry of conjugation within any split subgroup $H$ of the
full isometry group $G$ of $n$-dimensional Euclidean space. We prove that for
any $h \in H$, the conjugacy class $[h]_H$ of $h$ is described geometrically by
the move-set of its linearization, while the set of elements conjugating $h$ to
a given $h'\in [h]_H$ is described by the the fix-set of its linearization.
Examples include all affine Coxeter groups, certain crystallographic groups,
and the group $G$ itself.
