  We study the effect of discrete symmetry breaking in inhomogeneous scattering
media within the framework of generic wave propagation. Our focus is on
one-dimensional scattering potentials exhibiting local symmetries. We find a
class of spatially invariant nonlocal currents, emerging when the corresponding
generalized potential exhibits symmetries in arbitrary spatial domains. These
invariants characterize the wave propagation and provide a spatial mapping of
the wave function between any symmetry related domains. This generalizes the
Bloch and parity theorems for broken reflection and translational symmetries,
respectively. Their nonvanishing values indicate the symmetry breaking, whereas
a zero value denotes the restoration of the global symmetry where the
well-known forms of the two theorems are recovered. These invariants allow for
a systematic treatment of systems with any local symmetry combination,
providing a tool for the investigation of the scattering properties of
aperiodic but locally symmetric systems. To this aim we express the transfer
matrix of a locally symmetric potential unit via the corresponding invariants
and derive quantities characterizing the complete scattering device which serve
as key elements for the investigation of transmission spectra and particularly
of perfect transmission resonances.
