  Given a typical interval exchange transformation, we may naturally associate
to it an infinite sequence of matrices through Rauzy induction. These matrices
encode visitations of the induced interval exchange transformations within the
original. In 2010, W. A. Veech showed that these matrices suffice to recover
the original interval exchange transformation, unique up to topological
conjugacy, answering a question of A. Bufetov. In this work, we show that
interval exchange transformation may be recovered and is unique modulo
conjugacy when we instead only know consecutive products of these matrices.
This answers another question of A. Bufetov. We also extend this result to any
inductive scheme that produces square visitation matrices.
