  We introduce the concept of parity symmetry in restricted spatial domains --
local parity -- and explore its impact on the stationary transport properties
of generic, one-dimensional aperiodic potentials of compact support. It is
shown that, in each domain of local parity symmetry of the potential, there
exists an invariant quantity in the form of a non-local current, in addition to
the globally invariant probability current. For symmetrically incoming states,
both invariant currents vanish if weak commutation of the total local parity
operator with the Hamiltonian is established, leading to local parity
eigenstates. For asymmetrically incoming states which resonate within locally
symmetric potential units, the complete local parity symmetry of the
probability density is shown to be necessary and sufficient for the occurrence
of perfect transmission. We connect the presence of local parity symmetries on
different spatial scales to the occurrence of multiple perfectly transmitting
resonances and propose a construction scheme for the design of resonant
transparent aperiodic potentials. Our findings are illustrated through
application to the analytically tractable case of piecewise constant
potentials.
