Controlling light propagation across scattering media by wavefront shaping holds great promise for a wide range of communications and imaging applications. But, finding the right shape for the wavefront is a challenge when the mapping between input and output scattered wavefronts (that is, the transmission matrix) is not known. Correlations in transmission matrices, especially the so-called memory effect, have been exploited to address this limitation. However, the traditional memory effect applies to thin scattering layers at a distance from the target, which precludes its use within thick scattering media, such as fog and biological tissue. Here, we theoretically predict and experimentally verify new transmission matrix correlations within thick anisotropically scattering media, with important implications for biomedical imaging and adaptive optics.