We have measured both the differential travel times and the amplitude ratios of core phases PKP(DF) and PKP(BC), observed on digital broadband records worldwide, and analyzed them jointly, in order to try and distinguish between the effects of lateral heterogeneity acid anisotropy in the inner core, and to test different hypotheses on the possible origin of such anisotropy. For the best quality data, we observe a correlation between stronger attenuation and faster velocities on a global scale, confirming earlier results obtained beneath Africa. Regional variations are observed in this relationship. In addition, for paths which have their turning point in equatorial regions, both the velocity and attenuation show clear dependence on the angle Theta of the ray path in the inner core with the Earth's rotation axis. By contrast, for paths with turning points at higher latitudes, this dependence is apparent only in the travel times. The ray azimuth at its turning point, computed in a reference system with a slightly tilted symmetry axis with respect to the Earth's rotation, appears to be a more successful parameter for analyzing anisotropy. The velocity-attenuation correlation suggests that crystal orientation is responsible for both the travel time and attenuation anisotropy, and that the heterogeneity level inside the inner core must be small. However, in order to investigate the pattern of anisotropy in more detail, simple models expressed in terms of Theta alone are no longer adequate, and more complex parametrization, possibly related to convective processes, will have to be considered.