The splitting of normal modes sensitive to structure in the inner core is larger than can be explained by three-dimensional heterogeneity confined to the earth's mantle. A preferred interpretation of this anomalous splitting involves inner core anisotropy, providing a unified explanation of these data and of observed trends in the travel times of PKP waves that sample the inner core. We reexamine this interpretation, as well as a previously suggested alternative one in terms of outer core structure. Our motivation comes from recent results which indicate that simple, smooth models of inner core anisotropy are in disagreement with some PKP observations. We invert a recently assembled high quality dataset, comprising modes sensitive to mantle and outermost core structure, as well as modes sensitive to mantle, outer core and inner core structure. We compare models parametrized to include either inner core anisotropy or heterogeneity in the outer core. We show the following: (1) Outer core models, with fewer parameters, provide good overall fits to most modes with weak or strong sensitivity in the inner core, except for mode 3S2, the inner core mode with the strongest splitting. (2) Vs and &rgr; profiles obtained for models with outer core structure using mantle and core modes are in better agreement with those obtained using data sensitive to mantle structure alone, whereas the Vp structure in D″ shows a stronger c20 component when inner core anisotropy is considered. (3) Lateral heterogeneity restricted to the mantle, in particular strong heterogeneity in D″, fails to account consistently for the splitting of all modes. Outer core models thus deserve further investigation and should be weighted against more complex models of inner core anisotropy to explain anomalous splitting of normal modes and PKP travel times. The outer core structure obtained, while arguably unrealistic, is consistent with concentration of light elements, either confined to the Taylor cylinder tangent to the inner core or in polar caps at the top and bottom of the outer core. A small amount of shear within such polar caps might account for the excess splitting of some core modes. ¿ 2000 American Geophysical Union