We find simple expressions for the kinematic numerators of one-loop MHV amplitudes in maximally supersymmetric Yang-Mills theory and supergravity, at any multiplicity. The gauge-theory numerators satisfy the Bern-Carrasco-Johansson (BCJ) duality between color and kinematics, so that the gravity numerators are simply the square of the gauge-theory ones. The duality holds because the numerators can be written in terms of structure constants of a kinematic algebra, which is familiar from the BCJ organization of self-dual gauge theory and gravity. The close connection that we find between one-loop amplitudes in the self-dual case and in the maximally supersymmetric case is reminiscent of the dimension-shifting formula. The starting point for arriving at our expressions is the dimensional reduction of ten-dimensional amplitudes obtained in the field-theory limit of open superstrings.