  We study the statistical properties of stationary, isotropic and homogeneous
turbulence in two-dimensional (2D) flows, focusing on the direct cascade, that
is on wave-numbers large compared to the integral scale, where both energy and
enstrophy are provided to the fluid. Our starting point is the 2D Navier-Stokes
equation in the presence of a stochastic forcing, or more precisely the
associated field theory. We unveil two extended symmetries of the Navier-Stokes
action which were not identified yet, one related to time-dependent (or
time-gauged) shifts of the response fields and existing in both 2D and 3D, and
the other to time-gauged rotations and specific to 2D flows. We derive the
corresponding Ward identities, and exploit them within the non-perturbative
renormalization group formalism, and the large wave-number expansion scheme
developed in [Phys. Fluids {\bf 30}, 055102 (2018)]. We consider the flow
equation for a generalized $n$-point correlation function, and calculate its
leading order term in the large wave-number expansion. At this order, the
resulting flow equation can be closed exactly. We solve the fixed point
equation for the 2-point function, which yields its explicit time dependence,
for both small and large time delays in the stationary turbulent state. On the
other hand, at equal times, the leading order term vanishes, so we compute the
next-to-leading order term. We find that the flow equations for simultaneous
$n$-point correlation functions are not fully constrained by the set of
extended symmetries, and discuss the consequences.
