  The established diversity of electronic topology classes lends the
opportunity to pair them into dual topological complexes. Bulk-surface
correspondence then ensures the coexistence of a combination of boundary states
that cannot be realized but only at the various surfaces of such a dual
topological material. We show that the layered compound Bi_2TeI realizes a dual
topological insulator. It exhibits band inversions at two time reversal
symmetry points of the bulk band which classify it as a weak topological
insulator with metallic states on its (010) 'side' surfaces. Additional mirror
symmetry of the crystal structure concurrently classifies it as a topological
crystalline insulator. Bi2TeI is therefore predicted to host a pair of Dirac
cones protected by time reversal symmetry on its 'side' surfaces and three
pairs of Dirac cones protected by mirror symmetry on its 'top' and 'bottom'
(001) surfaces. We spectroscopically map the top cleaved surface of Bi_2TeI,
and crystallographic step edges therein. We show the existence of both two
dimensional surface states which are susceptible to mirror symmetry breaking,
as well as one dimensional channels that reside along the step edges. Their
mutual coexistence on the step edge where both facets join is facilitated by
momentum and energy segregation. Our observations of a dual topological
insulator make way to additional pairing of other dual topology classes with
distinct surface manifestations coexisting at their boundaries.
