  We revisit the problem of surface states in semiconductors with inverted band
structures, such as $\alpha$-Sn and HgTe. We unravel the confusion that arose
over the past decade regarding the origin of the surface states, their
topological nature, and the role of strain. Within a single minimalistic
description, we reconcile different solutions found in the 1980s with the
results obtained from modern-day numerical simulations, allowing us to
unambiguously identify all branches of surface states around the $\Gamma$-point
of the Brillouin zone in different regimes. We also show that strain is a
smooth "deformation" to the surface states, following the usual continuity
principle of physics, and not leading to any drastic change of the physical
properties in these materials, in contrast to what has recently been advanced
in the literature. We consider biaxial in-plane strain that is either tensile
or compressive, leading to different branches of surface states for topological
insulators and Dirac semimetals, respectively. Our model can help in
interpreting numerous experiments on topological surface states originating
from inverted-band semiconductors.
