  We study mirror symmetry of type II strings on manifolds with the exceptional
holonomy groups $G_2$ and Spin(7). Our central result is a construction of
mirrors of Spin(7) manifolds realized as generalized connected sums. In
parallel to twisted connected sum $G_2$ manifolds, mirrors of such Spin(7)
manifolds can be found by applying mirror symmetry to the pair of non-compact
manifolds they are glued from. To provide non-trivial checks for such geometric
mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds
in several new instances for both the Spin(7) and the $G_2$ case. For all of
these models we find possible assignments of discrete torsion phases, work out
the action of mirror symmetry, and confirm the consistency with the geometrical
construction. A novel feature appearing in the examples we analyse is the
possibility of frozen singularities.
