  We study the electronic surface states of the semiconducting alloy BiSb.
Using a phenomenological tight binding model we show that the Fermi surface of
the 111 surface states encloses an odd number of time reversal invariant
momenta (TRIM) in the surface Brillouin zone confirming that the alloy is a
strong topological insulator. We then develop general arguments which show that
spatial symmetries lead to additional topological structure, and further
constrain the surface band structure. Inversion symmetric crystals have 8 Z_2
"parity invariants", which include the 4 Z_2 invariants due to time reversal.
The extra invariants determine the "surface fermion parity", which specifies
which surface TRIM are enclosed by an odd number of electron or hole pockets.
We provide a simple proof of this result, which provides a direct link between
the surface states and the bulk parity eigenvalues. We then make specific
predictions for the surface state structure for several faces of BiSb. We next
show that mirror invariant band structures are characterized by an integer
"mirror Chern number", n_M. The sign of n_M in the topological insulator phase
of BiSb is related to a previously unexplored Z_2 parameter in the L point k.p
theory of pure Bi, which we refer to as the "mirror chirality", \eta. The value
of \eta predicted by the tight binding model for Bi disagrees with the value
predicted by a more fundamental pseudopotential calculation. This explains a
subtle disagreement between our tight binding surface state calculation and
previous first principles calculations on Bi. This suggests that the tight
binding parameters in the Liu Allen model of Bi need to be reconsidered.
Implications for existing and future ARPES experiments and spin polarized ARPES
experiments will be discussed.
