the left- and right-hand side in this expression and their global descendants contribute to $\Phi^{j\omega}_{J}(x|z)$ and $\Phi^{-1-j,\omega}_{J}(x|z)$, respectively. Moreover, Eq. (\[refletrucha\]) remains valid for any weight, as can be easily seen simply by acting with rising and lowering operators on both sides. Thus, by shifting to the $x$-basis we can write[^5] $$\label{refletrucha2}
\Phi^{j\omega}_{J}(x|z) = Y^{j\omega}_{J} \Phi^{-1-j,\omega}_{J}(x|z).$$ Note, however, that this $\mathbb Z_2$-symmetry does not constitute a “genuine” reflection symmetry in space-time. In particular, it does not generate any contact term for the propagator when acting on (\[regular\]), which is actually left invariant. In other words, Eq. (\[refletrucha2\]) is merely a remnant of the reflection symmetry in the unflowed frame.

As opposed to , a well-suited reflection should reduce to an integro-differential expression reducing to (\[intertwining\]) upon setting $\omega=0$. In the very same way that the intertwining operator adjusts the asymptotic behavior of $\Phi_{-1-j}(x|z)$ to that of $\Phi_{j}(x|z)$ in the H$^+_3$-WZNW model, the symmetry we seek should properly change the dependence of $\Phi^{-1-j,\omega}_{-1-J}(x|z)$ on the worldsheet coordinates to that of $\Phi^{j\omega}_{J}(x|z)$ as well. As for the Euclidean case, we furthermore expect an integration over the worldsheet with a power-law kernel, namely, an identity of the form $$\label{reflexion222}
\Phi^{j\omega}_{J}(x|z)=R^{j\omega}_J\left(\mathcal I^{j\omega}_J\Phi^{j\omega}_{-1-J}\right)(x|z),$$ with $$\label{refle1}
