Optical microresonators attract strong interest because of exciting effects and applications ranging from sensing of single atoms and molecules to quantum and nonlinear optics. For all this, control and tuning of the discrete resonances are vital. In resonators made of anisotropic materials that are beneficial for nonlinear-optical applications, anticrossings of ordinarily (o) and extraordinarily (e) polarized modes occur regularly. This effect is badly understood and harmful for mode control and tuning. We show that the anticrossings are inherent in the o- and e-modes because of the vectorial properties of Maxwell's equations. Within a novel pertubative approach employing a strong localization of the modes near the resonator rim, we have quantified the anticrossings. The values of avoidance gaps strongly exceeding the linewidths and selection rules for the interacting modes are predicted. The inferred values of the avoidance gaps are confirmed experimentally in resonators made of lithium niobate. Furthermore, based on theory, we have eliminated the anticrossings completely by spatially-controlled introduction of defects. This paves the way for unperturbed tuning of anisotropic microresonators.