The theory of fury developed turbulence is usually considered in an idealiz
ed homogeneous and isotropic state. Real turbulent flows exhibit the effect
s of anisotropic forcing. The analysis of correlation functions and structu
re functions in isotropic and anisotropic situations is facilitated and mad
e rational when performed in terms of the irreducible-representations of th
e relevant symmetry group which is the group of all rotations SO(3). In thi
s paper we first consider the needed general theory, and explain why we exp
ect different (universal) scaling exponents in the different sectors of the
symmetry group. We exemplify the theory context of isotropic turbulence (f
or third order tensorial structure functions) and in weakly anisotropic tur
bulence (for the second order structure function). The utility of the resul
ting expressions for the analysis of experimental data is demonstrated in t
he context of high Reynolds number measurements of turbulence in the atmosp
here.