We study the polarization-state formation in magnonic vortex crystals via scanning transmission x-ray microscopy. Self-organized state formation is observed by adiabatic reduction of a high-frequency field excitation. The emerging polarization patterns are shown to depend on the frequency of excitation and the strength of the dipolar interaction between the elements. In spite of the complexity of the investigated system, global order caused by local interactions creates polarization states with a high degree of symmetry. A fundamental dipole model and coupled equations of motion are adopted to analytically describe the experimental results. The emerging states can be predicted by a fundamental stability criterion based on the excitability of eigenmodes in the crystal. Micromagnetic simulations give additional insight into the underlying processes.