  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 extremal under-rotating limit it becomes
moduli-independent. We also discuss the issue of the fifth parameter
characterizing the four-dimensional seed solution, showing that it can be
generated by a transformation in the global symmetry group which is manifest in
the D=3 effective description.
