  There is considerable experimental evidence for the existence in Quantum Hall
systems of an approximate emergent discrete symmetry, $\Gamma_0(2) \subset
SL(2,Z)$. The evidence consists of the robustness of the tests of a suite a
predictions concerning the transitions between the phases of the system as
magnetic fields and temperatures are varied, which follow from the existence of
the symmetry alone. These include the universality of and quantum numbers of
the fixed points which occur in these transitions; selection rules governing
which phases may be related by transitions; and the semi-circular trajectories
in the Ohmic-Hall conductivity plane which are followed during the transitions.
We explore the implications of this symmetry for Quantum Hall systems involving
{\it two} charge-carrying fluids, and so obtain predictions both for bilayer
systems and for single-layer systems for which the Landau levels have a spin
degeneracy. We obtain similarly striking predictions which include the novel
new phases which are seen in these systems, as well as a prediction for
semicircle trajectories which are traversed by specific combinations of the
bilayer conductivities as magnetic fields are varied at low temperatures.
