A nearly perpendicular shock with a thickness on the order of electron inertial lengths has been reported recently by Newbury and Russell (Geophys. Res. Lett., vol. 23, pp. 781-784, 1996) based on the study of ISEE-1 and -2 crossings of Earth's bow shock. This paper examines the structure of nonlinear solitons and shocks propagating exactly perpendicular to the magnetic field based on the two-fluid model. It is shown that the characteristics of the solutions are distinct for three parameter regimes of the upstream plasma beta, β1, and magnetosonic number, Mms1. Magnetosonic solitons and purely resistive subcritical shocks with or without downstream magnetosonic wavetrains exist only for low β1 and Mms1. For subcritical shocks with intermediate β1 and Mms1, steady solutions exist only for resistivity greater than certain critical values. For subcritical shocks with smaller resistivity and for supercritical shocks (high β1 and Mms1), resistivity alone is insufficient to yield shock transitions. Including viscosity in the model calculations however leads to solutions of unstable upstream and downstream states. This chaotic nature may account for the highly turbulent structure of the observed thin shock.