Previous hypothesis concerning the cause of anomalous splitting in free oscillation spectra have led to models which are difficult to accept from the physical point of view-involving either substantial heterogeneity in the fluid outer core or large topographic variations in the core-mantle boundary and the inner core boundary and heterogeneity of several percent in inner core properties. Furthermore, other seismological evidence, some of it pre-existing and some of it very recently discovered, militates against these models. Within the framework of isotropic earth models there appears to be no acceptable explanation of the model observations. Here we show that anisotropy in the inner core can produce an effect which is of the correct magnitude and which varies from mode to mode in approximately the observed manner. The data currently available are insufficient to objectively map the anisotropy, but the simple assumption of a constant elastic tensor which is invariant under rotations about the Earth's rotation axis (i.e., is tranversely isotropic in the plane of the equator) matches well the gross features of the modal observations. This model does not entirely reconcile the modal data with travel-time observations; it is argued, however, that there exists an anisotropic model which will do so.