We study generalized two-field α-attractor models whose rescaled scalar manifold is the triply-punctured sphere endowed with its complete hyperbolic metric, whose underlying complex manifold is the modular curve Y(2). Using an explicit embedding into the end compactification, we compute solutions of the cosmological evolution equations for a few globally well-behaved scalar potentials, displaying particular trajectories with inflationary behavior as well as more general cosmological trajectories of surprising complexity. In such models, the orientation-preserving isometry group of the scalar manifold is isomorphic with the permutation group on three elements, acting on Y(2) as the group of anharmonic transformations. When the scalar potential is preserved by this action, α-attractor models of this type provide a geometric description of two-field “modular invariant j-models” in terms of gravity coupled to a non-linear sigma model with topologically non-trivial target and with a finite (as opposed to discrete but infinite) group of symmetries. The precise relation between the two perspectives is provided by the elliptic modular function λ, which can be viewed as a field redefinition that eliminates almost all of the countably infinite unphysical ambiguity present in the Poincaré half-plane description of such models.