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In QCD, we are used to describing the SU(3) color space in terms of a flow of color. At the algebra level, the Lorentz group consists of two copies of the (complexified) su(2) algebra, so one may anticipate that a similar way of thinking about the spacetime structure of scattering amplitudes should exist. In this article, we argue that this is indeed the case, and introduce the chirality-flow formalism for massless tree-level QED and QCD. Within the chirality-flow formalism, scattering amplitudes can directly be written down in terms of Lorentz-invariant spinor inner products, similar to how the color structure can be described in terms of a color flow. [ABSTRACT FROM AUTHOR]