The set \(\mathcal {Q}\) of reflections (i.e., operators C such that \(C^2=I\) ) in a \(\hbox {C}^*\) -algebra is a geometric space which has been the object of several investigations, and is an important tool in the study of these algebras. In this paper we consider a special class of reflections, the composition operators \(C_a\) acting on the Hardy space \(H^2\) of the unit disk, given by \(C_af=f\circ \varphi _a\) , where \(\begin{aligned} \varphi _a(z)=\frac{a-z}{1-{\bar{a}}z}, \end{aligned}\) for \(|a|<1\) . These operators are indeed reflections, because \(\varphi _a\circ \varphi _a=id\) . We study their eigenspaces \(N(C_a\pm I)\) , their relative position (i.e., the intersections between these spaces and their orthogonal complements for \(a\ne b\) in the unit disk) and the symmetries induced by \(C_a\) and these eigenspaces.
