  The phenomenological consequences of the residual Z^s_2 and Zbar^s_2
symmetries are explored in detail. With a precisely measured value of the
reactor angle, these two residual symmetries predict distinct distributions for
the Dirac CP phase and the atmospheric angle, which leads to the possibility of
identifying them at neutrino experiments. For both symmetries, it is possible
to resolve the neutrino mass hierarchy in most of the parameter space, and they
can be distinguished from one another if the true residual symmetry is Z^s_2
and the atmospheric angle is non-maximal. These results are obtained using an
equally split schedule: a 1.5-year run of neutrinos and a 1.5-year run of
antineutrinos at NOvA together with a 2.5-year run of neutrinos and a 2.5-year
run of antineutrinos at T2K. This schedule can significantly increase and
stabilize the sensitivities to the mass hierarchy and the octant of the
atmospheric angle with only a moderate compromise to the sensitivity of
distinguishing Z^s_2 and Zbar^s_2.
