  The 'double Dirac cone' 2D topological interface states found on the (001)
faces of topological crystalline insulators such as Pb$_{1-x}$Sn$_{x}$Se
feature degeneracies located away from time reversal invariant momenta, and are
a manifestation of both mirror symmetry protection and valley interactions.
Similar shifted degeneracies in 1D interface states have been highlighted as a
potential basis for a topological transistor, but realizing such a device will
require a detailed understanding of the intervalley physics involved. In
addition, the operation of this or similar devices outside of ultra-high vacuum
will require encapsulation, and the consequences of this for the topological
interface state must be understood. Here we address both topics for the case of
2D surface states using angle-resolved photoemission spectroscopy. We examine
bulk Pb$_{1-x}$Sn$_{x}$Se(001) crystals overgrown with PbSe, realizing
trivial/topological heterostructures. We demonstrate that the valley
interaction that splits the two Dirac cones at each $\bar{X}$ is extremely
sensitive to atomic-scale details of the surface, exhibiting non-monotonic
changes as PbSe deposition proceeds. This includes an apparent total collapse
of the splitting for sub-monolayer coverage, eliminating the Lifshitz
transition. For a large overlayer thickness we observe quantized PbSe states,
possibly reflecting a symmetry confinement mechanism at the buried topological
interface.
