After more than a century of history, the radiation-reaction problem in classical electrodynamics still surprises and puzzles new generations of researchers. Here we revise and explain some of the paradoxical issues that one faces when approaching the problem, mostly associated with regimes of uniform proper acceleration. The answers we provide can be found in the literature and are the synthesis of a large body of research. We just present them in a personal way that may help in their understanding. Besides, after the presentation of the standard answers we motivate and present a twist to those ideas. The physics of emission of radiation by extended charges (charges with internal structure) might proceed in a surprising oscillating fashion. This hypothetical process could open up new research paths and a new take on the equivalence principle.