Reflection-based QSP

Let $d\geq 0$. There exists a sequence of phase factors $\Phi=(\phi_0,\ldots,\phi_d)\in\mathbb{R}^{d+1}$ such that, for every $x\in[-1,1]$, $$ U_\Phi(x) = e^{i\phi_0 Z}\prod_{j=1}^{d}\left( \begin{bmatrix} x & -\sqrt{1-x^2}\\ \sqrt{1-x^2} & x \end{bmatrix}e^{i\phi_j Z}\right) = \begin{bmatrix} P(x) & -Q(x)\sqrt{1-x^2}\\ Q^*(x)\sqrt{1-x^2} & P^*(x) \end{bmatrix} $$ if and only if $P,Q\in\mathbb{C}[x]$ satisfy $\deg(P)\leq d$, $\deg(Q)\leq d-1$, $P$ has parity $d\bmod 2$, $Q$ has parity $(d-1)\bmod 2$, and $|P(x)|^2+(1-x^2)|Q(x)|^2=1$ for all $x\in[-1,1]$. At $d=0$, the degree condition on $Q$ means $Q=0$.