In recent experiments with ultracold gases a Raman coupling scheme is used to produce both spin-orbit (SO) and Zeeman-type couplings [Y.-J. Lin, Nature (London) 471, 83 (2011)NATUAS0028-083610.1038/nature09887]. Their competition drives a phase transition to a magnetized state with broken Z2 symmetry. Using a hydrodynamic approach we study a confined binary condensate subject to both SO and Zeeman-type couplings. We find that in the limit of small healing length and in the phase with unbroken symmetry, the boundary magnetization profile has an analytical solution in the form of a sine-Gordon soliton. The soliton is bound to the edge of the system by the nontrivial boundary condition resulting from the combined effect of the SO coupling and the drop in the particle density. The same boundary condition is important in the magnetized phase as well, where we characterize numerically the boundary spin structure. We further discuss how the nontrivial magnetization structure affects the density profile near the boundary, yet another prediction that can be tested in current experiments of spin-orbit-coupled condensates.