We use the combined action of 2-chiral reflections (T-dualities)
and shifts to build = 1,2 supersymmetric four-dimensional string
compactifications with few moduli. In particular, we consider
24 asymmetric orbifolds of Type IIB on the maximal
torus of SO(12) that mimic = 2
Calabi-Yau compactifications with small ``effective'' Hodge
numbers starting from (h11,h21) = (1,1). We analyze possible
... [Show full abstract] unoriented projections,
providing Type I examples with or
without open strings. = 1 oriented
asymmetric shift-orbifolds of Type IIB with few chiral multiplets are also presented.