The low-energy reaction 14C(n,gamma)15C provides a rare opportunity to test indirect methods for the determination of neutron capture cross sections by radioactive isotopes versus direct measurements. It is also important for various astrophysical scenarios. Currently, puzzling disagreements exist between the 14C(n,gamma)15C cross sections measured directly, determined indirectly, and calculated theoretically. To solve this puzzle, we offer a strong test based on a novel idea that the amplitudes for the virtual 15C-->14C + n and the real 15F -->14O + p decays are related. Our study of this relation, performed in a microscopic model, shows that existing direct and some indirect measurements strongly contradict charge symmetry in the 15C and 15F mirror pair. This brings into question the experimental determinations of the astrophysically important (n,gamma) cross sections for short-lived radioactive targets.