Let $s \in GL_2(\Bbb R)$ be a symmetric positive definite matrix (is this roughly the stretching part of the polar decomposition of some other matrix $x=sk$?). Conjugate $s$ by the reflection $$\gamma=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ Is there a $k_1 \in K = \left\{\begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}\right\}$, such that $\gamma s \gamma^{-1}=k_1 s k_1^{-1}$? How do I find it?
More generally, can we answer the question in the title?