Nonlinear phenomena and turbulence are central to our understanding and modeling of the dynamics of fluids and plasmas, and yet they still resist analytical resolution in many instances. However, progress has been made recently, displaying a richness of phenomena, which was somewhat unexpected a few years back, such as double constant-flux cascades of the same invariant for both large and small scales, or the presence of non-Gaussian wings in large-scale fields, for fluids and plasmas. Here, I will concentrate on the direct measurement of the magnitude of dissipation and the evaluation of intermittency in a turbulent plasma using exact laws stemming from invariance principles and involving cross-correlation tensors with both the velocity and the magnetic fields. I will illustrate these points through scaling laws, together with data analysis from existing experiments, observations and numerical simulations. Finally, I will also briefly explore the possible implications for the validity and use of several modeling strategies.