  We develop a folding approach to study two-dimensional symmetry-enriched
topological (SET) phases with the mirror reflection symmetry. Our folding
approach significantly transforms the mirror SETs, such that their properties
can be conveniently studied through previously known tools: (i) it maps the
nonlocal mirror symmetry to an onsite $\mathbb{Z}_2$ layer-exchange symmetry
after folding the SET along the mirror axis, so that we can gauge the symmetry;
(ii) it maps all mirror SET information into the boundary properties of the
folded system, so that they can be studied by the anyon condensation theory---a
general theory for studying gapped boundaries of topological orders; and (iii)
it makes the mirror anomalies explicitly exposed in the boundary properties,
i.e., strictly 2D SETs and those that can only live on the surface of a 3D
system can be easily distinguished through the folding approach. With the
folding approach, we derive a set of physical constraints on data that
describes mirror SET, namely mirror permutation and mirror symmetry
fractionalization on the anyon excitations in the topological order. We
conjecture that these constraints may be complete, in the sense that all
solutions are realizable in physical systems. Several examples are discussed to
justify this. Previously known general results on the classification and
anomalies are also reproduced through our approach.
