Employing the $mathbf{k}cdotmathbf{p}$ expansion for a family of tight-binding models for SmB$_6$, we analytically compute topological surface states on a generic $(lmn)$ surface. We show how the Dirac-cone spin structure depends on model ingredients and on the angle $theta$ between the surface normal and the main crystal axes. We apply the general theory to $(001)$, $(110)$, $(111)$, and $(210)$ surfaces, for which we provide concrete predictions for the spin pattern of surface states which we also compare with tight-binding results. As shown in previous work, the spin pattern on a $(001)$ surface can be related to the value of mirror Chern numbers, and we explore the possibility of topological phase transitions between states with different mirror Chern numbers and the associated change of the spin structure of surface states. Such transitions may be accessed by varying either the hybridization term in the Hamiltonian or the crystal-field splitting of the low-energy $f$ multiplets, and we compute corresponding phase diagrams.
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