We provide an exhaustive description of the magnetostatics of the uniformly polarized torus and its derivative self-intersecting (spindle) shapes. In the process, two complementary approaches have been implemented, position-space analysis of the Laplace equation with inhomogeneous boundary conditions and a Fourier-space analysis, starting from the determination of the shape amplitude of this topologically non-trivial body. The stray field and the demagnetization tensor have been determined as rapidly converging series of toroidal functions. The single independent demagnetization-tensor eigenvalue has been determined as a function of the unique aspect ratio α of the torus. Throughout the range of values of the ratio, corresponding to a multiply connected torus proper, the axial demagnetization factor Nz remains close to one half. There is no breach of smoothness of Nz(α) at the topological crossover to a simply connected tight torus (α=1). However, Nz is a non-monotonic function of the aspect ratio, such that substantially different pairs of corresponding tori would still have the same demagnetization factor. This property does not occur in a simply connected body of the same continuous axial symmetry. Several self-suggesting practical applications are outlined, deriving from the acquired quantitative insight.