  We study the bulk-boundary correspondence for topological crystalline phases,
where the crystalline symmetry is an order-two (anti)symmetry, unitary or
antiunitary. We obtain a formulation of the bulk-boundary correspondence in
terms of a subgroup sequence of the bulk classifying groups, which uniquely
determines the topological classification of the boundary states. This
formulation naturally includes higher-order topological phases as well as
topologically nontrivial bulk systems without topologically protected boundary
states. The complete bulk and boundary classification of higher-order
topological phases with an additional order-two symmetry or antisymmetry is
contained in this work.
