Various asymmetric orbifold models based on chiral shifts and chiral
reflections are investigated. Special attention is devoted to the consistency
of the models with two fundamental principles for asymmetric orbifolds :
modular invariance and the existence of a proper Hilbert space formulation for
states and operators. The interplay between these two principles is
non-trivial. It is shown, for example, that their simultaneous requirement
forces the order of a chiral reflection to be 4, instead of the naive 2. A
careful explicit construction is given of the associated one-loop partition
functions. At higher loops, the partition functions of asymmetric orbifolds are
built from the chiral blocks of associated symmetric orbifolds, whose pairings
are determined by degenerations to one-loop.Comment: 40 pages, no figures, typos correcte