  A bulk superconductor possessing a topological surface state at the Fermi
level is a promising system to realize long-sought topological
superconductivity. Although several candidate materials have been proposed,
experimental demonstrations concurrently exploring spin textures and
superconductivity at the surface have remained elusive. Here we perform
spectroscopic-imaging scanning tunnelling microscopy on the centrosymmetric
superconductor $\beta$-PdBi$_2$ that hosts a topological surface state. By
combining first-principles electronic-structure calculations and quasiparticle
interference experiments, we determine the spin textures at the surface, and
show not only the topological surface state but also all other surface bands
exhibit spin polarizations parallel to the surface. We find that the
superconducting gap fully opens in all the spin-polarized surface states. This
behaviour is consistent with a possible spin-triplet order parameter expected
for such in-plane spin textures, but the observed superconducting gap amplitude
is comparable to that of the bulk, suggesting that the spin-singlet component
is predominant in $\beta$-PdBi$_2$.
