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**Understanding Color-Kinematics Duality with a New Symmetry: From Radiation Zeros to BCJ**

I discuss a new set of symmetries obeyed by tree-level gauge-theory amplitudes involving at least one gluon. The symmetry acts as a momentum-dependent shift on the color factors of the amplitude. Using our previous development of radiation vertex expansions, we prove the invariance under this color-factor shift of the **n** -gluon amplitude, and in fact for any amplitudes involving at least one massless gauge boson and any number of massless or massive particles in arbitrary representations of the gauge group with spin zero, one-half, or one. The Bern-Carrasco-Johansson relations, which have already been proved for n-gluons and certain particle-gluon tree-level amplitudes with string theory methods and recursion relations, are a direct consequence of this symmetry. We also show that the amplitudes of the bi-adjoint scalar theory are invariant under the color-factor symmetry, and use this to derive the null eigenvectors of the propagator matrix. We generalize the color-factor shift to loop level, and prove the invariance under this shift of one-loop **n** -gluon amplitudes in any theory that admits a color-kinematic-dual representation of numerators. We show that the one-loop color-factor symmetry implies known relations among the integrands of one-loop color-ordered amplitudes.