Lie group symmetry in a mechanical system can lead to a dimensional reduction in its dynamical equations. Typically, the symmetries that one exploits are intrinsic to the mechanical system at hand, e. g., invariance of the system's Lagrangian to some group of motions. In the present work we consider symmetries that arise from an extrinsic control task, rather than the intrinsic structure of the configuration space, constraints, or system dynamics. We illustrate this technique with several examples. In the examples, the reduction enables us to design essentially global feedback controllers on the reduced systems. We also demonstrate how the proposed technique dovetails with Lagrangian reduction. We apply task-induced symmetry and reduction to a recently developed 6 DOF kinematic model of steerable bevel-tip needles. The resulting controllers cause the needle tip to track a subspace of its configuration space. We envision that the methodology presented in this paper will form the basis for a new planning and control framework for needle steering.