  We study the phase diagram of the inverted InAs/GaSb bilayer quantum wells.
For small tunneling amplitude between the layers, we find that the system is
prone to formation of an s-wave exciton condensate phase, where the
spin-structure of the order parameter is uniquely determined by the small
spin-orbit coupling arising from the bulk inversion asymmetry. The phase is
topologically trivial and does not support edge transport. On the contrary, for
large tunneling amplitude, we obtain a topologically non-trivial quantum spin
Hall insulator phase with a p-wave exciton order parameter, which enhances the
hybridization gap. These topologically distinct insulators are separated by an
insulating phase with spontaneously broken time-reversal symmetry. Close to the
phase transition between the quantum spin Hall and time-reversal broken phases,
the edge transport shows quantized conductance in small samples, whereas in
long samples the mean free path associated with the backscattering at the edge
is temperature independent, in agreement with recent experiments.
