Symmetries and reflections from composition operators in the disk

We study the composition operators $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_{a}f=f\circ {\phi }_a$, where $${\phi }_a(z)=\frac{a-z}{1-\bar{a}z},$$ for $\mathrm{\mid }a\mathrm{\mid }<1$. These operators are reflections: $C_a^2=1$. We study their eigenspaces $N(C_a\pm 1)$, their relative position (i.e., the intersections between these spaces and their orthogonal complementes for $a\mathrm{\ne }b$ in the unit disk) and the symmetries induced by $C_a$ and these eigenspaces.