How do we show $\lambda x . x (\lambda y .y) \equiv_{\alpha} \lambda y.y (\lambda x . x)$ in lambda calculus?
The two terms are in the same $\alpha$-equivalence class.
In lamba calculus, we can rename bound symbols without changing the meaning of the term.
$\lambda x.x$ and $\lambda y.y$ both mean "a function that takes one input and returns it". A variable is bound by the first matching lambda above it in the term-tree.
We often $\alpha$-rename bound variables before $\beta$-reduction to make sure unbound variables do not get caputured after the reduction step.



To answer your question:

$$\lambda x . x (\lambda y .y) \equiv_{\alpha} \lambda x.x (\lambda x'.x')$$ (rename bound $y$)
$$\equiv_{\alpha} \lambda y.y (\lambda x'.x')$$ (rename bound $x$)
$$\equiv_{\alpha} \lambda y.y (\lambda x.x)$$ (rename bound $x'$)
This gives us $\lambda x . x (\lambda y .y) \equiv_{\alpha} \lambda y.y (\lambda x.x)$