Deploying an analytical atomistic model of the bulk band structure of the IV-VI crystalline topological insulators, we connect the spin structure of the surface state to the crystal field and spin-orbit coupling parameters of the bulk material. While the Dirac-Weyl-type topological surface state is often assumed to be universal, we show that the physics of the surface state is strikingly nonuniversal, carrying a profound imprint of the bulk physics. To see this explicitly we calculate the Ruderman-Kittel-Kasuya-Yosida interaction, which may be viewed as a probe of this surface state spin structure, finding that its qualitative form depends on the values the bulk spin-orbit and crystal field parameters take. This opens the way to tune the spin interaction on the surface of a IV-VI topological insulator by, for instance, varying the composition of the IV-VI ternary compounds.