  We construct a theory of higher local symbols along Parsin chains for
reciprocity sheaves. Applying this formalism to differential forms, gives a new
construction of the Parsin-Lomadze residue maps, and applying it to the torsion
characters of the fundamental group gives back the reciprocity map from Kato's
higher local class field theory in the geometric case. The higher local symbols
satisfy various reciprocity laws. The main result of the paper is a
characterization of the modulus attached to a section of a reciprocity sheaf in
terms of the higher local symbols.
