  We introduce 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 the
radiation vertex expansion, we prove the invariance under this color-factor
shift of the $n$-gluon amplitude, as well as amplitudes involving massless or
massive particles in an arbitrary representation of the gauge group with spin
zero, one-half, or one. The Bern-Carrasco-Johansson relations are a direct
consequence of this symmetry.
  We also introduce the cubic vertex expansion of an amplitude, and use it to
derive a generalized-gauge-invariant constraint on the kinematic numerators of
the amplitude. We 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.
