The effective stress tensor of a homogeneous turbulent rotating fluid is anisotropic. This leads us to consider the most general axisymmetric four-rank "viscosity tensor" for a Newtonian fluid and the new terms in the turbulent effective force on large scales that arise from it, in addition to the microscopic viscous force. Some of these terms involve couplings to vorticity and others are angular momentum nonconserving (in the rotating frame). Furthermore, we explore the constraints on the response function and the two-point velocity correlation due to axisymmetry. Finally, we compare our viscosity tensor with other four-rank tensors defined in current approaches to nonrotating anisotropic turbulence.