We propose to discretize the convective and diffusive operators in the (incom- pressible) Navier-Stokes equations in such a manner that their symmetries are preserved. The resulting spatial discretization inherits all symmetry-related conservation and stability properties from the continuous formulation. In particularly, a symmetry-preserving discretization is unconditionally stable, and conserves the energy in the absense of viscous dissipation. In this paper, a fourth-order, symmetry-preserving discretization method is described and tested successfully for direct numerical simulation of turbulent channel flow.