  The most general construction of double affine Artin groups (DAAG) and Hecke
algebras (DAHA) associates such objects to pairs of compatible reductive group
data. We show that DAAG/DAHA always admit a faithful action by automorphisms of
a finite index subgroup of the Artin group of type $A_{2}$, which descends to a
faithful outer action of a congruence subgroup of $SL(2,\mathbb{Z})$ or
$PSL(2,\mathbb{Z})$. This was previously known only in some special cases and,
to the best of our knowledge, not even conjectured to hold in full generality.
  The structural intricacies of DAAG/DAHA are captured by the underlying
semisimple data and, to a large extent, by adjoint data; we prove our main
result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a
natural way to affine Lie algebras, or more precisely to their affinized Weyl
groups, which are the semi-direct products $W\ltimes Q^{\vee}$ of the Weyl
group $W$ with the coroot lattice $Q^{\vee}$. We now describe our results for
the adjoint case in greater detail.
  We first give a new Coxeter-type presentation for adjoint DAAG as quotients
of the Coxeter braid groups associated to certain crystallographic diagrams
that we call double affine Coxeter diagrams. As a consequence we show that the
rank two Artin groups of type $A_{2},B_{2},G_{2}$ act by automorphisms on the
adjoint DAAG/DAHA associated to affine Lie algebras of twist $r=1,2,3$,
respectively. This extends a fundamental result of Cherednik for $r=1$.
  We show further that the above rank two Artin group action descends to an
outer action of congruence subgroup $\Gamma_{1}(r)$. In particular $\Gamma_{1}(
r) $ acts naturally on the set of isomorphism classes of representations of an
adjoint DAAG/DAHA of twist type $r$, giving rise to a projective representation
of $\Gamma_{1}( r) $ on the space of a $\Gamma_{1}( r) $-stable representation.
