We study the intertwining relation XTφ=TψX where Tφ and Tψ are the Toeplitz operators induced on the Hardy space H2 by analytic functions φ and ψ, bounded on the open unit disc $\mathbb{U}$ , and X is a nonzero bounded linear operator on H2. Our work centers on the connection between intertwining and the image containment $\psi(\mathbb{U})\subset\varphi (\mathbb{U})$ , as well as on the nature of the intertwining operator X. We use our results to study the “extended eigenvalues” of analytic Toeplitz operators Tφ, i.e., the special case XTλφ=TφX, where λ is a complex number.