  The Homogeneous sine-Gordon (HSG) theories are integrable perturbations of
$G_k/U(1)^{r_G}$ coset CFTs, where $G$ is a simple compact Lie group of rank
$r_G$ and $k>1$ is an integer. Using their T-duality symmetries, we investigate
the relationship between the different theories corresponding to a given coset,
and between the different phases of a particular theory. Our results suggest
that for $G=SU(n)$ with $n\geq5$ and $E_6$ there could be two non-equivalent
HSG theories associated to the same coset, one of which has not been considered
so far.
