Two-dimensional thermal convection in the presence of a strong oblique magnetic field is studied using an asymptotic expansion in inverse powers of the Chandrasekhar number. The linear stability problem reveals the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length-scale whose vertical wavenumber kz is comparable with the large horizontal wavenumber k⊥ selected at onset, and a large-scale modulation which forms an envelope on the order of the layer depth d. The small-scale structure in the vertical results from magnetic alignment that forces fluid motions to be (nearly) parallel to the field lines. For convective transport in the vertical this constraint must be relaxed. This is achieved by molecular dissipation which allows weak upward (downward) motions of hot (cold) fluid elements across the field lines and results in a large-scale vertical modulation of the magnetic columns. Using the scaling suggested by the linear theory, strongly nonlinear steady and overstable solutions are constructed. These are characterized by large departures of the mean temperature profile from the conduction profile. For overstable rolls two modes of convection are uncovered. The first ‘vertical field’ mode is characterized by thin thermal boundary layers and a Nusselt number that increases rapidly with the applied Rayleigh number; this mode is typical of steady convection as well. The second or ‘horizontal field’ mode is present in overstable convection only and has broad thermal boundary layers and a Nusselt number that remains small and approximately independent of the Rayleigh number. At large Rayleigh numbers this regime is characterized by a piecewise linear temperature profile with a small isothermal core. The ‘horizontal field’ mode is favoured for substantial inclinations of the field and sufficiently small ohmic diffusivity. The transition between the two regimes is typically hysteretic and for fixed inclination and diffusivity may occur with increasing Rayleigh number. Similar but highly asymmetric states are obtained for depth-dependent ζ, where ζ is the ratio of ohmic to thermal diffusivity. These results are obtained from a nonlinear eigenvalue problem for the Nusselt number and mean temperature profile, and suggest a possible explanation for the sharp boundary between the umbra and penumbra in sunspots.