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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 5 and E 6 there could be two non-equivalent HSG theories associated with the same coset, one of which has not been considered so far.

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