Magnetic interactions underpin a plethora of magnetic states of matter, hence playing a central role both in fundamental physics and for future spintronic and quantum computation devices. The Dzyaloshinskii-Moriya interaction, D S S ij i j · ( ), being chiral and driven by relativistic effects, leads to the stabilization of highly-noncollinear spin textures such as skyrmions, which thanks to their topological nature are promising building blocks for magnetic data storage and processing elements.Here, we reveal and study a new chiral pair interaction, C S S S S ij i j i j · ( )( · ), which is the biquadratic equivalent of the DMI. First, we derive this interaction and its guiding principles from a microscopic model, and we connect the atomistic form to the micromagnetic one. Second, we study its properties in the simplest prototypical systems, magnetic 3d transition metal dimers deposited on the Pt(111), Pt(100), Ir(111), and Re(0001) surfaces, resorting to systematic first-principles calculations. Lastly, we discuss its importance and implications not only for magnetic dimers but also for extended systems, namely one-dimensional spin spirals and complex two-dimensional magnetic structures, such as a nanoskyrmion lattice found in an Fe monolayer on Ir(111).