$U_\omega$ acts on the plane $\mathrm{span}\{|s'\rangle, |\omega\rangle\}$ as the reflection across $|s'\rangle$

Let $f\colon\{0,\dots,N-1\}\to\{0,1\}$ have exactly one index $\omega$ with $f(\omega) = 1$, let $U_\omega|x\rangle = (-1)^{f(x)}|x\rangle$, and let $|s'\rangle = \frac{1}{\sqrt{N-1}}\sum_{x\ne\omega}|x\rangle$. Then for all $a, b \in \mathbb C$,

That is, on the plane spanned by $|s'\rangle$ and $|\omega\rangle$ the oracle is the reflection across $|s'\rangle$ (the reflection at the hyperplane orthogonal to $|\omega\rangle$); in particular it maps the plane to itself.

Formalization Note For $N = 1$ the sum defining $|s'\rangle$ is empty and $|s'\rangle = 0$; the statement then only concerns multiples of $|\omega\rangle$.