We explore thick accretion disks around rotating attractors. We detail the configurations analyzing the fluid angular momentum and finally provide a characterization of the disk morphology and different possible topologies. Investigating the properties of orbiting disks, a classification of attractors, possibly identifiable in terms of their spin-mass ratio, is introduced; then an attempt to characterize dynamically a series of different disk topologies is discussed, showing that some of the toroidal configuration features are determined by the ratio of the angular momentum of the orbiting matter and the spin-mass ratio of the attractor. Then we focus on "multistructured" disks constituted by two or more rings of matter orbiting the same attractor, andwe prove that some structures are constrained in the dimension of rings, spacing, location, and an upper limit of ring number is provided. Finally, assuming a polytropic equation of state, we study some specific cases.