On special flows over IETs that are not isomorphic to their inverses

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in \cite{Fr-Ku-Le}. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function.