  We consider two coupled time reversal invariant helical edge modes of the
same helicity, such as would occur on two stacked quantum spin Hall insulators.
In the presence of interaction, the low energy physics is described by two
collective modes, one corresponding to the total current flowing around the
edge and the other one describing relative fluctuations between the two edges.
We find that quite generically, the relative mode becomes gapped at low
temperatures, but only when tunneling between the two helical modes is
non-zero. There are two distinct possibilities for the gapped state depending
on the relative size of different interactions. If the intra-edge interaction
is stronger than the inter-edge interaction, the state is characterised as a
spin-nematic phase. However in the opposite limit, when the interaction between
the helical edge modes is strong compared to the interaction within each mode,
a spin-density wave forms, with emergent topological properties. Firstly, the
gap protects the conducting phase against localization by weak nonmagnetic
impurities; and secondly the protected phase hosts localized zero modes on ends
of the edge that may be created by sufficiently strong non-magnetic impurities.
