  We have investigated spin reorientation phenomena and interaction driven
effects under the presence of applied strains on the (001) surface of
Pb$_{1-x}$Sn$_x$(Te, Se) topological crystalline insulators, which host
multiple Dirac cones. Our analysis is based on a four-band $k\cdot p$ model,
which captures the spin and orbital textures of the surface states at low
energies around the $\bar{X}$ and $\bar{Y}$ points, including the Lifshitz
transition. Even without breaking the time-reversal symmetry, we find that
certain strains which break the mirror symmetry can induce hedgehog-like spin
texture associated with gap formation at the Dirac points. The Chern number of
the gapped surface ground state is shown to be tunable through the interplay of
strains and a perpendicular Zeeman field. We also consider effects of strain in
the presence of interactions in driving competing orders, and obtain the
associated phase diagram at the mean-field level. Potential applications of our
results for low power consuming electronics are discussed.
