Haar system rearrangements in Lorentz spaces

Let $\chi_n(t)$ $(n\ge 1)$ be Haar functions and let $\pi$ be a permutation of the set of natural numbers such that $\chi_{\pi(n)}(t)$ and $\chi_n(t)$ have supports of the same measure. We study the operators $T_\pi$ that are defined by the equalities $T_\pi\chi_n=\chi_{\pi(n)}$ $(n\ge 1)$. A criterion is found for boundedness of $T_\pi$ in the Lorentz spaces $L_{2,q}$. In particular, boundedness of $T_\pi$ in $L_{2,q}$ $(q\neq 2)$ implies that $T_\pi$ is an isomorphism of $L_p$ onto itself for all $p\in(1,\infty)$.