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 $S: \mathbb{T}_1 \to \mathbb{T}_1$ by the formula \begin{align} S(t)= t+c_j \quad \text{mod} \, 1 , \quad t \in [t_j,t_{j+1}) \end{align} Here, the $c_j$ are real values. Each interval has its own $c_j$. They could be either the same or different from each other. Such map is called interval translation map (ITM) or, if it is one-to-one it is called interval exchange map (IEM).
It is easy to construct an example of an Interval Exchange Map, for example, the rotation map is a good example of that, but I think it is kind of hard to construct an example of an ITM which is not an IEM. Could anyone help me find a simple example?
Choose $n=1$ and write $\mathbb T^1 \simeq [0,1) = [0,1/2) \cup [1/2,1) $. Then, define \begin{align} S(t)= \begin{cases} t + 1/4 & t \in [0,1/2) \\ t-1/4 & t \in [1/2,1) \end{cases} \end{align} Unless I misunderstood your definitions, this seems to work.