Seeking conceptual explanation of these nice bijections on roots of unity 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?
 A: The question and the analog to the Cayley map in complex numbers pointed out in the comment by Jyrki Lahtonen is not only an analog, but in fact both are special cases of this more general observation: Let $z\mapsto\bar z$ be an involutory automorphism of a field $F$, and let $E$ be the subfield fixed by $\bar{\phantom{a}}$. Furthermore, set $S=\{z\in F\;|\;z\bar z=1\}$. Then, for $b\in S$, $c\in F\setminus E$, and $d\in F\setminus S$,
\begin{equation*}
z\mapsto \frac{cz-b\bar c}{z-b}
\end{equation*}
maps $S$ to $E\cup\{\infty\}$ and
\begin{equation*}
z\mapsto \frac{z-b\bar d}{dz-b}
\end{equation*}
maps $S$ to $S$.
A: Let $E$ be a curve defined by a singular Weierstrass equation over $\mathbb{F}_q$, where the singularity is a node, say at the origin. Then, Silverman says (Arithmetic of Elliptic Curves, page 46) that $E$ may be written as
$$
E: y^2 + A_1 xy - A_2 x^2 - x^3 = 0,
$$
where $A_1^2 + 4 A_2$ is not zero.
If the two tangent lines at the node are given by $y = a_i x$ for $i=1,2$, then by comparing coefficients we see that the slopes are roots of $t^2 + A_1 t - A_2$.
Suppose this quadratic is irreducible over $\mathbb{F}_q$. Then it splits in $L = \mathbb{F}_{q^2}$. According to Silverman's Prop. 2.5 on page 56 (properly adjusted to not assume algebraic closure, see Exercise 3.5 on page 105), the map $(x,y) \rightarrow (y - a_1 x)/(y - a_2 x)$ gives us an isomorphism of algebraic groups $E_{ns}(L)\cong L^\times$. Part (ii) of the exercise referenced above shows that under this isomorphism, $E_{ns}(K)$ corresponds precisely to those elements of $L^\times$ whose norm to $K$ is $1$. Now $N(x) = x\cdot x^q = x^{q+1}$, so elements of norm $1$ are precisely the $(q+1)th$ roots of unity in $\mathbb{F}_{q^2}$. Since $E$ is singular with a point defined over $\mathbb{F}_q$, its non-singular part is isomorphic to $\mathbb{P}^1(\mathbb{F}_q)$.
So, we have a bijection between the $(q+1)th$ roots of unity in $\mathbb{F}_{q^2}$ and $\mathbb{P}^1(\mathbb{F}_q)$ given by a linear fractional transformation.
