  We study the low-energy surface electronic structure of the transition-metal
dichalcogenide superconductor PdTe$_2$ by spin- and angle-resolved
photoemission, scanning tunneling microscopy, and density-functional
theory-based supercell calculations. Comparing PdTe$_2$ with its sister
compound PtSe$_2$, we demonstrate how enhanced inter-layer hopping in the
Te-based material drives a band inversion within the anti-bonding p-orbital
manifold well above the Fermi level. We show how this mediates spin-polarised
topological surface states which form rich multi-valley Fermi surfaces with
complex spin textures. Scanning tunneling spectroscopy reveals type-II
superconductivity at the surface, and moreover shows no evidence for an
unconventional component of its superconducting order parameter, despite the
presence of topological surface states.
