  We introduce a new concept of interval rearrangement ensembles (IRE), which
is a generalization of interval exchange transformations (IET). This
construction expands the space of IETs in accordance with the natural duality
that we pinpoint. Induction of Rauzy\,--\,Veech kind is applicable to IREs. It
is conjugate to the reverse operation by the duality mentioned above. A natural
extension of an IRE is associated with two transversal flows on a flat
translation surface with branching points.
