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Supermassive black hole binaries (SMBHBs) are natural by-products of galaxy mergers and are expected to be powerful multi-messenger sources. They can be powered by the accretion of matter and then radiate across the electromagnetic spectrum, much like normal active galactic nuclei (AGNs). Current electromagnetic observatories have a good chance of detecting and identifying these systems in the near future. However, precise observational indicators are needed to distinguish individual AGNs from SMBHBs. In this paper, we propose a novel electromagnetic signature from SMBHBs: non-thermal emission produced by the interaction between the jets ejected by the black holes. We study close SMBHBs, which accrete matter from a circumbinary disc and the mini-discs formed around each hole. Each black hole ejects a magnetically dominated jet in the direction of its spin through the Blandford--Znajeck mechanism. We argue that in such a situation, the interaction between the jets can trigger strong magnetic reconnection events, where particles are accelerated and emit non-thermal radiation. Depending on whether the jets are aligned or misaligned, this radiation can have different periodicities. We model the evolution of the particles accelerated during the dual jet interaction and calculate their radiative output, obtaining spectra and providing estimates for the variability timescales. We finally discuss how this emission compares with that of normal AGNs., Comment: 11 pages, 5 figures. Accepted to be published in MNRAS