We write a gravity theory with Yang–Mills-type action using the biconformal gauging of the conformal group. We show that the resulting biconformal Yang–Mills gravity theories describe 4-dim, scale-invariant general relativity in the case of slowly changing fields. In addition, we systematically extend arbitrary 4-dim Yang–Mills theories to biconformal space, providing a new arena for studying flat-space Yang–Mills theories. By applying the biconformal extension to a 4-dim pure Yang–Mills theory with conformal symmetry, we establish a 1-1, onto mapping between a set of gravitational gauge theories and 4-dim, flat-space gauge theories.