Summary: Robust estimation of location for data on the unit sphere \(\mathcal{S}^{p-1}\) is an important problem in directional statistics even though the influence functions of the sample mean direction and other location estimators are bounded. A significant limitation of previous literature on this topic is that robust estimators and procedures have been developed under the assumption that the underlying population is rotationally symmetric. This assumption often does not hold with real data and in these cases there is a needless loss of efficiency in the estimator. In this article, we propose two estimators for spherical data, both of which are analogous to the affine-equivariant spatial median in Euclidean space. The influence functions of the new location estimators are obtained under a new semiparametric elliptical symmetry model on the sphere and are shown to be standardized bias robust in the highly concentrated case; the influence function of the companion scatter matrix is also obtained. An iterative algorithm that computes both estimators is described. Asymptotic results, including consistency and asymptotic normality, are also derived for the location estimators that result from applying a fixed number of steps in this algorithm. Numerical studies demonstrate that both location estimators may be expected to perform well in practice in terms of efficiency and robustness. A brief example application from the geophysics literature is also provided.