A geometrical approach to a radial intermediary model for an axisymmetric rigid body in roto-orbital motion is presented. The presence of symmetries enables a well-suited formulation by choosing action-angle type variables. Singularities associated with the angles are avoided by introducing extra fictitious variables and performing a symplectic transformation leading to a global, quaternionic double-chart. Then, making use of the SO(3) and T-4 symmetry of our model, a full reduction process by stages is carried out, which in combination with the constrained dynamics related to the fictitious variables, leads to a 1-DOF reduced-constrained system. Our program includes a parametric analysis of relative equilibria and a complete description of the fibers in the reconstruction of the reduced system. (C) 2020 Elsevier B.V. All rights reserved.