  Experiments studying renormalization group flows in the quantum Hall system
provide significant evidence for the existence of an emergent holomorphic
modular symmetry $\Gamma_0(2)$. We briefly review this evidence and show that,
for the lowest temperatures, the experimental determination of the position of
the quantum critical points agrees to the parts \emph{per mille} level with the
prediction from $\Gamma_0(2)$. We present evidence that experiments giving
results that deviate substantially from the symmetry predictions are not cold
enough to be in the quantum critical domain. We show how the modular symmetry
extended by a non-holomorphic particle-hole duality leads to an extensive web
of dualities related to those in plateau-insulator transitions, and we derive a
formula relating dual pairs $(B,B_d)$ of magnetic field strengths across any
transition. The experimental data obtained for the transition studied so far is
in excellent agreement with the duality relations following from this emergent
symmetry, and rule out the duality rule derived from the ``law of corresponding
states". Comparing these generalized duality predictions with future
experiments on other transitions should provide stringent tests of modular
duality deep in the non-linear domain far from the quantum critical points.
