The phenomenon of ionospheric cross-modulation occurs when a 'wanted' radio wave passes through a region of the ionospheric plasma where a powerful modulated 'disturbing' radio wave is strongly absorbed. Many previous studies have assumed that the wanted wave is vertically incident and, in some cases, that the ionosphere is isotropic. The theory is studied here for an obliquely incident wanted wave propagating through an anisotropic ionosphere. The Booker quartic equation is used to find the modulation transferred to the wanted wave, and how it depends on height, angle of incidence, azimuth and radio frequency. It is assumed that the disturbing wave has a sinusoidal amplitude modulation so that the collision frequency in the disturbed region varies periodically with the modulation frequency. The physical processes that occur in various situations are reviewed. It is found that many of the results from the simpler theories still apply, but some new effects are found. For example, there can be a marked difference in the modulation transferred to two ordinary (or two extraordinary) wanted waves with the same angle of incidence, one travelling obliquely upwards, and the other downwards through the disturbed region. It is found that an increase in the average power of the disturbing wave does not necessarily imply a corresponding increase in the modulation transferred to the wanted wave. The radio engineer who requires an accurate assessment of cross-modulation as a possible source of interference in communications could apply some methods of this paper with the aid of modern computers.