In this paper we consider the local controllability problem for control-affine systems that are homogeneous with respect to a one-parameter family of dilations corresponding to time-scaling in the control. We construct and derive properties of a variational cone that completely characterizes local controllability for these homogeneous systems. In the process, we are able to give a bound on the order, in terms of the integers describing the dilation, of perturbations that do not alter the local controllability property. Our approach uses elementary Taylor expansions and avoids unnecessarily complicated open mapping theorems to prove local controllability. Examples are given that illustrate the main results.