We present a detailed mass reconstruction and a novel study on the substructure properties in the core of the Cluster Lensing And Supernova survey with Hubble (CLASH) and Frontier Fields galaxy clusterMACS J0416.1-2403. We reproduce the measured positions of a set of 30 multiple images with a remarkable median offset of only 0.3 by means of a comprehensive strong lensing model comprised of two cluster dark-matter halos, represented by cored elliptical pseudo-isothermal mass distributions, and the cluster member components, parameterized with dual pseudo-isothermal total mass profiles. We investigate the galaxy contribution, both in terms of total and stellar mass, to the total mass budget of the cluster. Our findings of the location and shape of the cluster dark-matter halo density profiles and on the cluster substructures provide intriguing tests of the assumed collisionless, cold nature of dark matter and of the role played by baryons in the process of structure formation.