What decomposition of $[0,1]$ would get rid of the point $0$? I feel like the idea would involve picking a point $u\in [0,1]$ and sending $[0,u]$ to $[1-u,1]$ but that would require sending $(u, 1]$ to $(0,1-u]$ but this decomposition is not disjoint.
Here, two sets $A, B$ are equidecomposable, if there exist finite disjoint decompositions $A=A_1 \cup A_2\cup \dots\cup A_n$ and $B=B_1 \cup B_2 \cup\dots \cup B_n$ such that $A_i$ and $B_i$ are isometric for $1\leq i\leq n$.