What is the name of the operator that translates from $X\rightarrow(Y\rightarrow Z)$ to $Y\rightarrow(X\rightarrow Z)$?
The function $$\lambda f.\lambda x.\lambda y.f\;y\;x$$ of type $$\forall X. \forall Y. \forall Z.(X \to Y \to Z) \to Y \to X \to Z$$ is often called flip. This is the case in Haskell (see here), and in some OCaml libraries as well (see here). According to wikipedia, people call this function (or combinator) $C$ in the context of combinatory logic (that name sounds pretty random though).