The imaging of proteins within macromolecular complexes has been limited by the low axial resolution of optical microscopes. To overcome this problem, we propose a novel computational reconstruction method that yields isotropic resolution in fluorescence imaging. The guiding principle is to reconstruct a single volume from the observations of multiple rotated particles. Our new operational framework detects particles, estimates their orientation, and reconstructs the final volume. The main challenge comes from the absence of initial template and a priori knowledge about the orientations. We formulate the estimation as a blind inverse problem, and propose a block-coordinate stochastic approach to solve the associated non-convex optimization problem. The reconstruction is performed jointly in multiple channels. We demonstrate that our method is able to reconstruct volumes with 3D isotropic resolution on simulated data. We also perform isotropic reconstructions from real experimental data of doubly labeled purified human centrioles. Our approach revealed the precise localization of the centriolar protein Cep63 around the centriole microtubule barrel. Overall, our method offers new perspectives for applications in biology that require the isotropic mapping of proteins within macromolecular assemblies.