Leading models relate short gamma-ray bursts (SGRBs) to a relativistic jet launched by the black hole (BH)-accretion torus system that can be formed in a binary neutron star (BNS) or a NS-BH binary merger. However, recent observations by Swift have revealed a large fraction of SGRB events accompanied by X-ray afterglows with durations ~10^2-10^5 s, suggesting continuous energy injection from a long-lived central engine, which is incompatible with the short (<1 s) accretion timescale of a BH-torus system. The formation of a supramassive NS (SMNS), resisting the collapse on much longer spin-down timescales, can explain these afterglow durations, but leaves serious doubts on whether a relativistic jet can be launched at merger. Here we present a novel scenario that can solve this dichotomy, in which the SGRB is produced after the eventual collapse of the SMNS, but observed before (part of) its long-lasting spin-down emission. The "time reversal" in the observation of the two signals is caused by the substantial delay affecting the spin-down emission, due to the optically thick environment surrounding the system generated by the early differential rotation and the subsequent spin-down emission itself.