  An element $f$ of a group $G$ is reversible if it is conjugated in $G$ to its
own inverse; when the conjugating map is an involution, $f$ is called strongly
reversible. We describe reversible maps in certain groups of interval exchange
transformations namely $G_n \simeq (\mathbb S^1)^n \rtimes\mathcal S_n $, where
$\mathbb S^1$ is the circle and $\mathcal S_n $ is the group of permutations of
$\{1,...,n\}$. We first characterize strongly reversible maps, then we show
that reversible elements are strongly reversible. As a corollary, we obtain
that composites of involutions in $G_n$ are product of at most four
involutions. We prove that any reversible Interval Exchange Transformation
(IET) is reversible by a finite order element and then it is the product of two
periodic IETs. In the course of proving this statement, we classify the free
actions of $BS(1,-1)$ by IET and we extend this classification to free actions
of finitely generated torsion free groups containing a copy of $\mathbb Z^2$.
We also give examples of faithful free actions of $BS(1,-1)$ and other groups
containing reversible IETs. We show that periodic IETs are product of at most
$2$ involutions. For IETs that are products of involutions, we show that such
3-IETs are periodic and then are product of at most $2$ involutions and we
exhibit a family of non periodic 4-IETs for which we prove that this number is
at least $3$ and at most $6$.
