Abstract Gamma-ray bursts (GRBs), both long and short, are explosive events whose inner engine is generally expected to be a black hole or a highly magnetic neutron star (magnetar) accreting high-density matter. Recognizing the nature of GRB central engines, and in particular the formation of neutron stars (NSs), is of high astrophysical significance. A possible signature of NSs in GRBs is the presence of a plateau in the early X-ray afterglow. Here we carefully select a subset of long and short GRBs with a clear plateau, and look for an additional NS signature in their prompt emission, namely a transition between the accretion and propeller phases in analogy with accreting, magnetic compact objects in other astrophysical sources. We estimate from the prompt emission the minimum accretion luminosity below which the propeller mechanism sets in, and the NS magnetic field and spin period from the plateau. We demonstrate that these three quantities obey the same universal relation in GRBs as in other accreting compact objects switching from accretion to propeller. This relation provides also an estimate of the radiative efficiency of GRBs, which we find to be several times lower than radiatively efficient accretion in X-ray binaries and in agreement with theoretical expectations. These results provide additional support to the idea that at least some GRBs are powered by magnetars surrounded by an accretion disk.