While clusters of galaxies are regarded as one of the most important cosmological probes, the conventional spherical modelling of the intracluster medium and the dark matter (DM), and the assumption of strict hydrostatic equilibrium (i.e. the equilibrium gas pressure is provided entirely by thermal pressure) are very approximate at best. Extending our previous works, we developed further a method to reconstruct for the first time the full 3D structure (triaxial shape and principal-axis orientation) of both DM and intracluster (IC) gas, and the level of non-thermal pressure of the IC gas. We outline an application of our method to the galaxy cluster Abell 383, taken as part of the Cluster Lensing and Supernova Survey with Hubble (CLASH) multicycle treasury programme, presenting results of a joint analysis of X-ray and strong lensing measurements. We find that the intermediate–major and minor–major axis ratios of the DM are 0.71 ± 0.10 and 0.55 ± 0.06, respectively, and the major axis of the DM halo is inclined with respect to the line of sight of 211 ± 101. The level of non-thermal pressure has been evaluated to be about 10 per cent of the total energy budget. We discuss the implications of our method for the viability of the cold dark matter (CDM) scenario, focusing on the concentration parameter C and the inner slope of the DM, γ, since the cuspiness of DM density profiles in the central regions is one of the critical tests of the CDM paradigm for structure formation: we measure γ= 1.02 ± 0.06 on scales down to 25 Kpc, and C= 4.76 ± 0.51, values which are close to the predictions of the standard model, and providing further evidences that support the CDM scenario. Our method allows us to recover the 3D physical properties of clusters in a bias-free way, overcoming the limitations of the standard spherical modelling and enhancing the use of clusters as more precise cosmological probes.