I proved the following facts by unenlightening calculations. Since the statements are quite clean, I think there should be a conceptual explanation for them, which my proof certainly is not.

Let $q$ be a prime power, and let $\mu_{q+1}$ be the set of $(q+1)$-th roots of unity in the finite field $\mathbf{F}_{q^2}$. If $b\in\mu_{q+1}$ and $c\in\mathbf{F}_{q^2}\setminus\mathbf{F}_q$ then $$ x\mapsto \frac{cx-bc^q}{x-b} $$ maps $\mu_{q+1}$ to $\mathbf{F}_q\cup\{\infty\}$. If $b\in\mu_{q+1}$ and $d\in\mathbf{F}_{q^2}\setminus\mu_{q+1}$ then $$ x\mapsto \frac{x-bd^q}{dx-b} $$ maps $\mu_{q+1}$ to itself. (It is also true that these are the only degree-one rational functions which map $\mu_{q+1}$ to either $\mathbf{F}_q\cup\{\infty\}$ or $\mu_{q+1}$, but I'm mainly interested in understanding the existence.)

I tagged this "group theory" because the first fact vaguely feels like a connection between orbits of a nonsplit torus and a split torus in $\textrm{PGL}_2(q)$. It's tempting to identify $\mathbf{F}_{q^2}$ with $\mathbf{F}_q\times\mathbf{F}_q$, and consider the resulting action of $\textrm{GL}_2(q)$ on $\mathbf{F}_{q^2}$, but I don't see how to go further in this way.

Any suggestions?