  We develop a formalism relating nonlocal current continuity to spatial
symmetries of subparts in discrete Schr\"odinger systems. Breaking of such
local symmetries hereby generates sources or sinks for the associated nonlocal
currents. The framework is applied to locally inversion-(time-) and
translation-(time-) symmetric one-dimensional photonic waveguide arrays with
Hermitian or non-Hermitian effective tight-binding Hamiltonians. For stationary
states the nonlocal currents become translationally invariant within symmetric
domains, exposing different types of local symmetry. They are further employed
to derive a mapping between wave amplitudes of symmetry-related sites,
generalizing also the global Bloch and parity mapping to local symmetry in
discrete systems. In scattering setups, perfectly transmitting states are
characterized by aligned invariant currents in attached symmetry domains, whose
vanishing signifies a correspondingly symmetric density. For periodically
driven arrays, the invariance of the nonlocal currents is retained on period
average for quasi-energy eigenstates. The proposed theory of symmetry-induced
continuity and local invariants may contribute to the understanding of wave
structure and response in systems with localized spatial order.
