How many vectors can one construct by by reflecting a vector $b\in\mathbb{R}^d$ for $b\neq 0$?
Reflections can be described by Householder matrices $H=I-2vv^T/||v||_2^2$.
In other words, I'm interested in the subset $S(b):=\{v\mid \exists \text{ Householder Matrix } H: v=Hb\}\subseteq \mathbb{R}^d$.
Is it true that $S(b)=\{ v \mid ||v||_2^2=||b||_2^2\}$?
Yep! Every reflection of $b$ has the same norm, as $H$ is orthogonal. Specifically, \begin{align*} H^\top H &= \left(I - 2\frac{vv^\top}{\|v\|^2}\right)^\top \left(I - 2\frac{vv^\top}{\|v\|^2}\right) \\ &= \left(I - 2\frac{vv^\top}{\|v\|^2}\right) \left(I - 2\frac{vv^\top}{\|v\|^2}\right) \\ &= I - 4\frac{vv^\top}{\|v\|^2} + 4\frac{v(v^\top v) v^\top}{\|v\|^4} \\ &= I - 4\frac{vv^\top}{\|v\|^2} + 4\frac{v v^\top}{\|v\|^2} \\ &= I = HH^\top. \end{align*} Therefore $\|Hb\| = \|b\|$ for all such matrices $H$.
On the other hand, choose any $c$ such that $\|c\| = \|b\|$, and let $v = b - c$. I claim that the $H$ corresponding to $v$ maps $b$ to $c$. We have \begin{align*} Hb - c &= \left(I - 2\frac{vv^\top}{\|v\|^2}\right)b - c \\ &= b - 2\frac{vv^\top b}{\|v\|^2} - c \\ &= b - c - 2(v \cdot b) \frac{v}{\|v\|^2} \\ &= b - c - 2((b - c) \cdot b) \frac{(b - c)}{\|b - c\|^2} \\ &= ((b - c) \cdot (b - c) - 2(b - c) \cdot b)\frac{b - c}{\|b - c\|^2} \\ &= ((b - c) \cdot (-b - c))\frac{b - c}{\|b - c\|^2} \\ &= (c \cdot c - b \cdot b)\frac{b - c}{\|b - c\|^2} \\ &= (\|c\|^2 - \|b\|^2)\frac{b - c}{\|b - c\|^2} \\ &= 0. \end{align*}