Interval Translation maps :
Consider the circle $\mathbb{T}^1= \frac{\mathbb{R}}{\mathbb{Z}}$. We represent it as a union of disjoint subsegments $M_j = [t_j; t_{j+1})$, $j = 0, \cdots, n, t_0 = t_n$ and define the map $S$ by the formula \begin{align} S(t)=t+c_j, \quad \text{mod}1, \quad t\in M_j \end{align} Here $c_j$ are real values. Such map is called interval translation (ITM).
Interval Exchange Maps:
In some recent paper its definition is :
Any one-to-one ITM is called IEM.
The original definition made by Michael Keane (1975):
Let $X = [0,1) $ and $n \geq 2$ is an integer. For each probability vector $\alpha =(\alpha_1 \cdots \alpha_n )$ with $\alpha_i >0$ for $1 \leq i \leq n$, we set: $\beta_0 =0$ and $\beta_i = \sum\limits_{j=1}^{i}\alpha_j$ and $X_i=[\beta_{i-1}, \beta_{i})$. Let $\tau$ be a permutation of the symbols ${1, ... , n}$ then $\alpha^{\tau}=(\alpha_{\tau^{-1}(1)} , ..., \alpha_{\tau^{-1}(n)} )$ is a probability vector with positive components, and we can form the corresponding $\beta_{i}^{\tau}$ and $X_{i}^{\tau}$: $\beta_{i}^{\tau}=\sum\limits_{j=1}^{i}\alpha_{\tau^{-1}(j)}$ and $X_{i}^{\tau}=[\beta_{i-1}^{\tau},\beta_{i}^{\tau} )$. Now define : $S: [0,1) \to [0,1)$ where $S(x)= x - \beta_{i-1} + \beta_{\tau(i)-1}^{\tau}$ for each $x \in X_i$. Such map is called Interval Exchange Map or IEM.
Now I want to see why these two different definitions of IEM are equivalent?
Actually I need to see why for every one to one ITM there is a permutation like $\tau$ such that the original definition is concluded.
Note that in the second definition of ITM the map $S$ takes $X_i$ to $X^\tau_{\tau(i)}$: this follows immediately from $$ S(x)= x - \beta_{i-1} + \beta_{\tau(i)-1}^{\tau}. $$ That is, $$S(X_i)=X^\tau_{\tau(i)}\quad\text{and so}\quad S(X_{\tau^{-1}(i)})=X^\tau_{i}.$$
So, to answer your question, take an IEM that is an ITM according to the first definition. To find an appropriate $\tau$ you just need to consider the images of $X_1,\ldots, X_n$, thus $$S(X_1),\ldots, S(X_n),$$ then order them successively on the circle, say $$S(X_{\tau^{-1}(1)})=X^\tau_1,\ldots, S(X_{\tau^{-1}(n)})=X^\tau_n$$ for some permutation $\tau$. This will be precisely your desired permutation.