 Solution

Choose orthonormal coordinates with origin $P$ and horizontal axis $l$. The Euclidean reflection and planar rotation have matricesTheir function composition therefore satisfies the composition of a plane rotation and a reflection identityConjugating by a planar rotation transports the horizontal reflection axis to the line at angle $\alpha /2$. Equivalently, the vector $e=(\cos (\alpha /2),\sin (\alpha /2))$ is fixed, whereas its perpendicular $e_{\perp }=(-\sin (\alpha /2),\cos (\alpha /2))$ is sent to $-e_{\perp }$. Consequently the fixed line is the line through $P$ obtained by rotating $l$ through $\alpha /2$, and $\tau \rho$ is reflection in that line. Angles of an unoriented line are understood modulo $\pi$, so this description is independent of the representative chosen for $\alpha$ modulo $2\pi$.