**Question:** Given a quadratic irrational $x = a + b\sqrt{D}$ ($a,b \in \Bbb{Q}$, $D \in \Bbb{N}_{> 0}$ square-free) and its Galois conjugate $x' = a - b\sqrt{D}$, is it true that the continued fraction expansions of $x$ and $x'$ have the same period?

Computations of a few random example seems to suggest that is indeed the case, and I think I even saw a statement to this effect somewhere, but can't remember where.

**Motivation**: The continued fraction expansions of two numbers $x,x' \in \Bbb{R}$ conjugate via an element of $\text{PGL}_2(\Bbb{Z})$ are 'almost the same', i.e. $a_{k+n} = a'_{k+m}$ for some $n,m$ and all $k \geq 0$, and I suspect that in the case of two conjugate quadratic-irrationals there always exists a 'generalized reflection' $s_H = wsw^{-1}$, $w \in \text{PGL}_2(\Bbb{Z})$, $s \in \left\{\begin{bmatrix} & 1 \\ 1& \end{bmatrix}, \begin{bmatrix} -1 & 1 \\ & 1\end{bmatrix}, \begin{bmatrix} -1 & \\ & 1\end{bmatrix}\right\}$ such that $s_H(x) = x'$.