With reference to the effective three-dimensional description of stationary, single center solutions to (ungauged) symmetric supergravities, we complete a previous analysis on the definition of a general geometrical mechanism for connecting global symmetry orbits (duality orbits) of non-extremal solutions to those of extremal black holes. We focus our attention on a generic representative of these orbits, providing its explicit description in terms of D = 4 fields. As a byproduct, using a new characterization of the angular momentum in terms of quantities intrinsic to the geometry of the D = 3 effective model, we are able to prove on general grounds its invariance, as a function of the boundary data, under the D = 4 global symmetry. In the ...