Rotating nuclear matter is defined as the system of infinitely many nucleons in a rotating frame neglecting the electrostatic interaction and centrifugal single-nucleon potential. We study the ground state of this system as a function of the densities of neutrons and protons. In the limit where the angular velocity is much smaller than the Fermi energy, the structure of the single-nucleon density corresponds to anisotropic spin distributions at the surfaces of local neutron and proton Fermi spheres. The anisotropy results from the non-central terms in the effective two-nucleon interaction. Contrary to the situation in a system of non-interacting nucleons, the spin asymmetry induced by rotation is a strongly non-linear function of the Fermi momentum. In symmetric nuclear matter at normal density it equals roughly that of the non-interacting system due to mutually cancelling contributions from the spin-orbit and central parts of the effective two-nucleon interaction. The volume contributions to the moments of inertia and single-nucleon Routhian of finite nuclei are calculated, and estimates obtained of certain surface contributions to the moment of inertia.