  Let IET be the group of bijections from $\mathopen{[}0,1 \mathclose{[}$ to
itself that are continuous outside a finite set, right-continuous and piecewise
translations. The abelianization homomorphism $f: \text{IET} \to A$, called
SAF-homomorphism, was described by Arnoux-Fathi and Sah. The abelian group $A$
is the second exterior power of the reals over the rationals.
  For every subgroup $\Gamma$ of $\mathbb{R/Z}$ we define $\text{IET}(\Gamma)$
as the subgroup of $\text{IET}$ consisting of all elements $f$ such that $f$ is
continuous outside $\Gamma$. Let $\tilde{\Gamma}$ be the preimage of $\Gamma$
in $\mathbb{R}$. We establish an isomorphism between the abelianization of
$\text{IET}(\Gamma)$ and the second skew-symmetric power of $\tilde{\Gamma}$
over $\mathbb{Z}$ denoted by ${}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}}
\tilde{\Gamma}$. This group often has non-trivial $2$-torsion, which is not
detected by the SAF-homomorphism.
  We then define $\text{IET}^{\bowtie}$ the group of all interval exchange
transformations with flips. Arnoux proved that this group is simple thus
perfect. However for every subgroup $\text{IET}^{\bowtie}(\Gamma)$ we establish
an isomorphism between its abelianization and $\langle \lbrace a \otimes a ~
[\text{mod}~2] \mid a \in \tilde{\Gamma} \rbrace \rangle \times \langle \lbrace
\ell \wedge \ell ~ [\text{mod}~2] \mid \ell \in \tilde{\Gamma} \rbrace \rangle$
which is a $2$-elementary abelian subgroup of $\bigotimes^2_{\mathbb{Z}}
\tilde{\Gamma} / (2\bigotimes^2_{\mathbb{Z}} \tilde{\Gamma}) \times
{}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tilde{\Gamma} / (2
{}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tilde{\Gamma})$.
