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The interfacial Dzyaloshinskii-Moriya interaction (DMI) holds promises for design and control of chiral spin textures in low-dimensional magnets with efficient current-driven dynamics. Recently, an interlayer DMI has been found to exist across magnetic multilayers with a heavy-metal spacer between magnetic layers. This opens the possibility of chirality in these three-dimensional magnetic structures. Here we show the existence of the interlayer DMI in a synthetic antiferromagnetic multilayer with both inversion and in-plane asymmetry. We analyse the interlayer DMI's effects on the magnetization and the current-induced spin-orbit torque (SOT) switching of magnetization through a combination of experimental and numerical studies. The chiral nature of the interlayer DMI leads to an asymmetric SOT switching of magnetization under an in-plane magnetic field. Our work paves the way for further explorations on controlling chiral magnetizations across magnetic multilayers through SOTs, which can provide a new path in the design of SOT devices. Exchange interactions between spins in a magnetic system can be engineered using magnetic multilayer systems in order to investigate a rich variety of magnetic phenomenon. Here the authors investigate the nature and control of magnetization of a synthetic antiferromagnetic stack with built in planar asymmetry and find evidence of an asymmetric current-driven switching in the chiral magnetization across the magnetic multilayers. [ABSTRACT FROM AUTHOR]