Let \(C_g:f\mapsto f\circ g\) be the composition operator on the Hardy space \(H^2(\Omega)\), where \(\Omega\) is either the unit disc \(\mathbb D\) or the right half-plane \(\mathbb C_+\), and \(g\) is such that \(C_g\) is isometric; the latter means that \(g\) is an inner function and \(g(0)=0\), for \(\Omega=\mathbb D\), and that \(\Phi=M\circ g\circ M\), where \(M(z)=\frac{1-z} {1+z}\), is~an inner function such that \(\frac{1+\Phi(z)}{1+z}\) belongs to \(H^2\) and has unit norm, for \(\Omega=\mathbb C_+\). The~authors show that the invariant subspace lattice of such operators \(C_g\) is very rich, by~constructing invariant subspaces \(\mathcal M\) such that \(C_g| \mathcal M\) is a bilateral shift of infinite multiplicity (for \(\Omega=\mathbb C_+\) and \(\deg\Phi=1\)) or a unilateral shift of any given multiplicity~\(N\), \(1\leq N\leq\infty\) (for \(\Omega=\mathbb C_+\) and \(\deg\Phi>1\), or~\(\Omega=\mathbb D\) and \(\deg g>1\)). They also characterize invariant subspaces of \(C_g\) of the form \(BH^2\), with \(B\) a Blaschke product, as~well as common invariant subspaces of \(C_g\) and the shift semigroup \(S_\tau: f(z)\mapsto e^{-z\tau} f(z)\), \(\tau\geq0\), on~\(H^2(\mathbb C_+)\).