# What determines helicity of 3D topological insulator surface state

Consider a 3D topological insulator surface state Hamiltonian with mass possibly due to magnetic doping $$\sum_{i,j=x,y}v_{ij} k_i \sigma_j + m\sigma_z.$$ What determines the helicity $$X=\mathrm{sgn}[\det(v_{ij})]$$ and the sign of mass? Are they tunable or do they vary among different systems or opposite surfaces of one system? I usually see the form $$p_x\sigma_y-p_y\sigma_x$$ with $$X=1$$. Is it the same for all cases and why?