Conjugacy classes of the orthogonal group $O(2)$ I was inspired by this question conjugacy classes of the special orthogonal group $SO(2)$, but in that case the answer is simple bacause $SO(2)$ is abelian. However, $O(2)$ is not, so how can we describe the structure of the conjugacy classes of this group?
 A: All reflections in lines are conjugate (by suitable rotations).  Rotations through the same angle in opposite directions are conjugate (by any reflections). To see that the preceding two sentences are the whole story, use that conjugation preserves determinants and traces of matrices.
A: Background
By computing the determinant of the defining condition $AA^T=I$ of the group $O(2)$ we see that the determinant of an orthogonal matrix is either $+1$ or $-1$. This shows that $SO(2)$ is a subgroup of $O(2)$ with index two.
In two dimensions, we can use the condition that the columns of an orthogonal matrix must be unit norm orthogonal vectors, we can obtain the following parametrization of orthogonal matrices with determinant $+1$
$$
R(\theta)=\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}
$$
and the following parametrization of orthogonal matrices with determinant $-1$
$$
R(\theta)Z=\begin{bmatrix}
\cos\theta & \sin\theta \\
\sin\theta & -\cos\theta
\end{bmatrix}
$$
where $Z=\mathrm{diag}(1, -1)$. The matrices of the form $R(\theta)$ describe rotations and the matrices of the form $R(\theta)Z$ describe rotoreflections.
Note that $R(\theta)Z=ZR(-\theta)$ for any angle $\theta$.
Conjugacy classes
Conjugacy class $\mathrm{Cl}(R(\alpha))$ of a rotation $R(\alpha)$ is
$$
\begin{align}
\mathrm{Cl}(R(\alpha))&=\{R(\theta)R(\alpha)R(-\theta)\,|\,\theta\in[0, 2\pi)\}\cup\{R(\theta)ZR(\alpha)ZR(-\theta)\,|\,\theta\in[0, 2\pi)\} \\
&= \{R(\alpha)\}\cup\{R(-\alpha)\} \\
&= \{R(\alpha),R(-\alpha)\}.
\end{align}
$$
Conjugacy class of a rotoreflection $R(\alpha)Z$ is
$$
\begin{align}
\mathrm{Cl}(R(\alpha)Z)&=\{R(\theta)R(\alpha)ZR(-\theta)\,|\,\theta\in[0, 2\pi)\}\cup\{R(\theta)ZR(\alpha)ZZR(-\theta)\,|\,\theta\in[0, 2\pi)\} \\
&= \{R(\theta)Z\,|\,\theta\in[0,2\pi)\}\cup\{R(\theta)Z\,|\,\theta\in[0,2\pi)\} \\
&= \{R(\theta)Z\,|\,\theta\in[0,2\pi)\}.
\end{align}
$$
We conclude that every rotation is conjugate to itself and its inverse and all rotoreflections are conjugate to each other.
