The cold diffuse interstellar medium (ISM) is turbulent with supersonic, most likely transAlfvénic motions. All its tracers, including velocity, are structured down to the smallest scales accessible to observations. This structure, though, is not random and can be quantified using the statistics of its Fourier phase increments. The quantity of structure of interstellar clouds is larger than that of fractal Brownian motion (fBm) fields and of mildly compressible turbulence. We propose that this structure is a response of the medium to a general property of turbulence, the space-time intermittency of its dissipation. The bursts of turbulent dissipation, because they temporarily (over a few hundred years) and locally heat the diffuse medium up to ~103 K, irreversibly modify the subsequent thermal and chemical evolution of the gas. Preliminary models of such bursts predict chemical abundances and excitation of a subset of molecules that are in agreement with the observations. Timescales of the velocity fluctuations are short enough to locally decouple the neutrals from the ionized species in the diffuse medium and generate helical magnetic fields. Small scale fluctuations of the polarization properties of the diffuse ISM are therefore expected.