Given an interval exchange transformation (IET) and a sub-interval, there arises a natural visitation matrix that relates the induced IET to the original IET. We show that the original IET, up to topological conjugacy, may be recovered from successive visitation matrices. This answers a question by A. Bufetov and generalizes work by W. A. Veech, which considered the case when the matrices arise from Rauzy induction. Furthermore, we provide an effective proof of Veech's result. That is to say, we will show how to find the necessary data for an IET given only a finite number of such visitation matrices.