  Although piecewise isometries (PWIs) are higher dimensional generalizations
of one dimensional interval exchange transformations (IETs), their generic
dynamical properties seem to be quite different. In this paper we consider
embeddings of IET dynamics into PWI with a view to better understanding their
similarities and differences. We derive some necessary conditions for existence
of such embeddings using combinatorial, topological and measure theoretic
properties of IETs. In particular, we prove that continuous embeddings of
minimal $2$-IETs into orientation preserving PWIs are necessarily trivial and
that any $3$-PWI has at most one non-trivially continuously embedded minimal
$3$-IET with the same underlying permutation. Finally, we introduce a family of
$4$-PWIs with apparent abundance of invariant nonsmooth fractal curves
supporting IETs, that limit to a trivial embedding of an IET.
