From pg. 35 of Lambda Calculus and Combinators An Introduction:
Corollary 3.3.1 in $\lambda$ and $CL$: for every $Z$ and $n \ge 0$, the equation
$$ xy_1 \ldots y_n = Z $$
can be solved for $x$. That is, there is a term $X$ such that
$$ Xy_1 \ldots y_n =_{\beta, w} [X/x]Z. $$
Proof: Choose $X \equiv \mathsf{Y} (\lambda x y_1 \ldots y_n.Z)$.
Note: In the book, we assume that $\mathsf{Y} \equiv (\lambda ux. xuux)(\lambda ux. xuux)$. Further, for ease of notation, we can let $V \equiv (\lambda x y_1 \ldots y_n.Z)$.
As can be seen from this question, I am having trouble replicating the proof of this corollary on its own. That being said, even assuming it is true and that I understood why, I am having trouble understanding one of its consequences. In particular, the author later asserts that
[This corollary] provides the world with ... a bureaucrat $X_2$ which eternally permutes its arguments with no other effect,
$$ X_2 y z =_{\beta, w} X_2 z y $$
But why is this?
Attempting to use the Corollary, we can set $z = x$ and $Z = X_2 y z$ and observe that it is true that
$$ Xy_1 \ldots y_n =_{\beta, w} [X/x] X_2 yx $$
for some $X$.