A current challenge in the field of magnetoelectric multiferroics is to identify systems that allow a controlled tuning of states displaying distinct magnetoelectric responses. Here we show that the multiferroic ground state of the archetypal multiferroic TbMnO₃ is dramatically modified by epitaxial strain. Neutron diffraction reveals that in highly strained films the magnetic order changes from the bulk-like incommensurate bc-cycloidal structure to commensurate magnetic order. Concomitant with the modification of the magnetic ground state, optical second-harmonic generation (SHG) and electric measurements show an enormous increase of the ferroelectric polarization, and a change in its direction from along the c- to the a-axis. Our results suggest that the drastic change of multiferroic properties results from a switch of the spin-current magnetoelectric coupling in bulk TbMnO₃ to symmetric magnetostriction in epitaxially-strained TbMnO₃. These findings experimentally demonstrate that epitaxial strain can be used to control single-phase spin-driven multiferroic states.