When choosing between options, we must solve an important binding problem. The values of the options must be associated with information about the action needed to select them. We hypothesize that the brain solves this binding problem through use of distinct population subspaces. To test this hypothesis, we examined the responses of single neurons in five reward-sensitive regions in rhesus macaques performing a risky choice task. In all areas, neurons encoded the value of the offers presented on both the left and the right side of the display in semi-orthogonal subspaces, which served to bind the values of the two offers to their positions in space. Supporting the idea that this orthogonalization is functionally meaningful, we observed a session-to-session covariation between choice behavior and the orthogonalization of the two value subspaces: trials with less orthogonalized subspaces were associated with greater likelihood of choosing the less valued option. Further inspection revealed that these semi-orthogonal subspaces arose from a combination of linear and nonlinear mixed selectivity in the neural population. We show this combination of selectivity balances reliable binding with an ability to generalize value across different spatial locations. These results support the hypothesis that semi-orthogonal subspaces support reliable binding, which is essential to flexible behavior in the face of multiple options.