We study a generalization \(\operatorname {Rec}_d\) of the group \(\operatorname {IET}=\operatorname {Rec}_1\) of interval exchange transformations in every dimension \(d \ge 1\) , called the rectangle exchange transformations group. The subset of restricted rotations in \(\operatorname {IET}\) is a generating subset and we prove that a natural generalization of these elements, called restricted shuffles, form a generating subset of \(\operatorname {Rec}_d\) . We denote by \(\mathscr {T}_d\) the subset of \(\operatorname {Rec}_d\) made up of those transformations that permute two disjoint rectangles by translations. We prove that the derived subgroup of \(\operatorname {Rec}_d\) is generated by \(\mathscr {T}_d\) . We also identify the abelianization of \(\operatorname {Rec}_d\) .
