Many Sun-like stars are observed to host close-in super-Earths (SEs) as part of a multiplanetary system. In such a system, the spin of the SE evolves due to spin–orbit resonances and tidal dissipation. In the absence of tides, the planet’s obliquity can evolve chaotically to large values. However, for close-in SEs, tidal dissipation is significant and suppresses the chaos, instead driving the spin into various steady states. We find that the attracting steady states of the SE’s spin are more numerous than previously thought, due to the discovery of a new class of ‘mixed-mode’ high-obliquity equilibria. These new equilibria arise due to subharmonic responses of the parametrically driven planetary spin, an unusual phenomenon that arises in non-linear systems. Many SEs should therefore have significant obliquities, with potentially large impacts on the physical conditions of their surfaces and atmospheres.