Various model-independent aspects of the $\overline{K}N\to K\Xi$ reaction are investigated, starting from the determination of the most general structure of the reaction amplitude for $\Xi$ baryons with $J^P={\frac{1}{2}}^{\pm }$ and ${\frac{3}{2}}^{\pm }$ and the observables that allow a complete determination of these amplitudes. Polarization observables are constructed in terms of spin-density matrix elements. Reflection symmetry about the reaction plane is exploited, in particular, to determine the parity of the produced $\Xi$ in a model-independent way. In addition, extending the work of Biagi et al. [Z. Phys. C 34, 175 (1987)], a way of determining simultaneously the spin and parity of the ground state of $\Xi$ baryons as well as those of the excited $\Xi$ states is presented.