Let $Y\to X$ be a Galois cover of complex varieties (or only consider complex algebraic curves) with the ramification data $(p_\bullet,\eta_\bullet)$, namely branch points $p_\bullet$ with a preferred generator $\eta_i$ of the stabilizer $G_{g_i}$ for each $p_i$. (To be precise, the preferred generator is the one corresponding to the monodromy of the simple counter-clock circle around the branch point.)
We know that, there is another Galois cover $Y’\to X$ with the reverse ramification data $(p_\bullet,\eta_\bullet^{-1})$.
My question is: is there any nature $X$-morphism between these two Galois covers? If so, What can we say about it?
More generally, what if we take the ramification data $(p_\bullet,\eta_\bullet^{m_\bullet})$ (where $m_\bullet$ are integers)?