Anisotropic nanophotonic structures can couple the levels of a quantum emitter through the quantum interference effect. In this paper we study the coupling of quantum emitter excited states through the modes of a fully anisotropic structure, a structure for which all directions are physically nonequivalent. We consider an anisotropic metasurface as an illustrative example of such a structure. We point out another degree of freedom in controlling the temporal dynamics and spectral profiles of quantum emitters; namely, we show that a combination of the metasurface anisotropy and tilt of the emitter quantization axis with respect to the metasurface normal results in nonsymmetric dynamics between the transitions of electrons from the left-circular state to the right-circular state and in the inverse process. Our findings give an additional mechanism for control over light emission by quantum systems and can be utilized for probing active transitions of quantum emitters.