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After being launched, GRB jets propagate through dense media prior to their breakout. The jet-medium interaction results in the formation of a complex structured outflow, often referred to as a "structured jet". The underlying physics of the jet-medium interaction that sets the post-breakout jet morphology has never been explored systematically. Here we use a suite of 3D simulations to follow the evolution of hydrodynamic long and short gamma-ray bursts (GRBs) jets after breakout to study the post-breakout structure induced by the interaction. Our simulations feature Rayleigh-Taylor fingers that grow from the cocoon into the jet, mix cocoon with jet material and destabilize the jet. The mixing gives rise to a previously unidentified region sheathing the jet from the cocoon, which we denote the jet-cocoon interface (JCI). long GRBs undergo strong mixing, resulting in most of the jet energy to drift into the JCI, while in short GRBs weaker mixing is possible, leading to a comparable amount of energy in the two components. Remarkably, the jet structure (jet-core plus JCI) can be characterized by simple universal angular power-law distributions, with power-law indices that depend solely on the mixing level. This result supports the commonly used power-law angular distribution, and disfavors Gaussian jets. At larger angles, where the cocoon dominates, the structure is more complex. The mixing shapes the prompt emission light curve and implies that typical long GRB afterglows are different from those of short GRBs. Our predictions can be used to infer jet characteristics from prompt and afterglow observations.