  Symmetric non-expanding horizons are studied in arbitrary dimension. The
global properties -as the zeros of infinitesimal symmetries- are analyzed
particularly carefully. For the class of NEH geometries admitting helical
symmetry a quasi-local analog of Hawking's rigidity theorem is formulated and
proved: the presence of helical symmetry implies the presence of two
symmetries: null, and cyclic.
  The results valid for arbitrary-dimensional horizons are next applied in a
complete classification of symmetric NEHs in 4-dimensional space-times (the
existence of a 2-sphere crossection is assumed). That classification divides
possible NEH geometries into classes labeled by two numbers - the dimensions
of, respectively, the group of isometries induced in the horizon base space and
the group of null symmetries of the horizon.
