 Orthogonal involutions of the plane

For a real orthogonal involution, $A^{-1}=A=A^T$, so it is a symmetric matrix. Its eigenvalues are $\pm 1$. In two dimensions the possibilities are the identity, the half-turn $-I$, and reflectionsThe reflection fixes the line making angle $\theta /2$ with the first coordinate axis and negates its perpendicular direction. This follows either from the two corresponding eigenvectors or by rotating $\mathrm{diag}(1,-1)$. A nonidentity element of order two in a faithful representation must be either the half-turn or a reflection; the identity matrix is excluded by faithfulness.