  A three-dimensional strong-topological-insulator or -semimetal hosts
topological surface states which are often said to be gapless so long as
time-reversal symmetry is preserved. This narrative can be mistaken when
surface state degeneracies occur away from time-reversal-invariant momenta. The
mirror-invariance of the system then becomes essential in protecting the
existence of a surface Fermi surface. Here we show that such a case exists in
the strong-topological-semimetal Bi$_4$Se$_3$. Angle-resolved photoemission
spectroscopy and \textit{ab initio} calculations reveal partial gapping of
surface bands on the Bi$_2$Se$_3$-termination of Bi$_4$Se$_3$(111), where an 85
meV gap along $\bar{\Gamma}\bar{K}$ closes to zero toward the mirror-invariant
$\bar{\Gamma}\bar{M}$ azimuth. The gap opening is attributed to an interband
spin-orbit interaction that mixes states of opposite spin-helicity.
