We study Kondo physics of a spin-$frac{1}{2}$ impurity in electronic matter with strong spin-orbit interaction, which can be realized by depositing magnetic adatoms on the surface of a three-dimensional topological insulator. We show that magnetic properties of topological surface states and the very existence of Kondo screening strongly depend on details of the bulk material, and specifics of surface preparation encoded in time-reversal preserving boundary conditions for electronic wavefunctions. When this tunable Kondo effect occurs, the impurity spin is screened by purely orbital motion of surface electrons. This mechanism gives rise to a transverse magnetic response of the surface metal, and spin textures that can be used to experimentally probe signatures of a Kondo resonance. Our predictions are particularly relevant for STM measurements in ${rm Pb Te}$-class crystalline topological insulators, but we also discuss implications for other classes of topological materials.
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