A <maximal torus> is $T=SO_2$, which is normal of index two in $O_2$. Therefore
$$
\boxed{W=O_2/SO_2\cong C_2}.
$$
A reflection represents its nontrivial element and conjugates a rotation of angle $\theta$ to the rotation of angle $-\theta$.

Solution

Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / A / 3 / b / ii / Solution

 Solution (source code)