The paper is concerned with the spectral theory of composition operators on the Hardy space \(H^2\) of the unit disk. The author proves that, under certain conditions on the symbol \(\varphi \) (involving in particular fixed points), the point-spectrum of the adjoint \(C_\varphi^*\) of \(C_\varphi\) contains a disk centered at the origin. It is also shown that under these conditions the associated eigenspaces are infinite dimensional. Also studied are weighted composition operators. The proofs are based on \textit{P. Poggi-Corradini}'s work [Rev. Mat. Iberoam. 19, No. 3, 943--970 (2003; Zbl 1057.30023)] on the existence of backward iteration sequences for \(\varphi\). Unveiled are \(C_\varphi^*\)-invariant subspaces \(M\) such that \(C^*_\varphi|_M\) is similar to a backward weighted shift.