  Characters of $E_8\otimes E_8$ and SO(32) heterotic strings involving the
full internal symmetry Cartan subalgebra generators are defined after circle
compactification so that they are T dual. The novel point, as compared with an
earlier study of the type II case (hep-th/9707107), is the appearence of Wilson
lines. Using SO(17,1) transformations between the weight lattices reveals the
existence of an intermediate theory where T duality transformations are
disentangled from the internal symmetry. This intermediate theory corresponds
to a sort of twisted compactification of a novel type. Its modular invariance
follows from an interesting interplay between three representations of the
modular group.
