  To obtain a unified framework for symmetric and asymmetric heterotic orbifold
constructions we provide a systematic study of Narain compactifications
orbifolded by finite order T-duality subgroups. We review the generalized
vielbein that parametrizes the Narain moduli space (i.e. the metric, the
B-field and the Wilson lines) and introduce a convenient basis of generators of
the heterotic T-duality group. Using this we generalize the space group
description of orbifolds to Narain orbifolds. This yields a unified,
crystallographic description of the orbifold twists, shifts as well as Narain
moduli. In particular, we derive a character formula that counts the number of
unfixed Narain moduli after orbifolding. Moreover, we develop new machinery
that may ultimately open up the possibility for a full classification of Narain
orbifolds. This is done by generalizing the geometrical concepts of Q-, Z- and
affine classes from the theory of crystallography to the Narain case. Finally,
we give a variety of examples illustrating various aspects of Narain orbifolds,
including novel T-folds.
