  The present paper aims at elaborating a completely physical representation
for the general 4-parameter family of the extended double-Kerr spacetimes
describing two spinning sources in gravitational equilibrium. This involved
problem is solved in a concise analytical form by using the individual Komar
masses and angular momenta as arbitrary parameters, and the simplest
equatorially symmetric specialization of the general expressions obtained by us
yields the physical representation for the well-known Dietz-Hoenselaers
superextreme case of two balancing identical Kerr constituents. The existence
of the physically meaningful "black hole-superextreme object" equilibrium
configurations permitted by the general solution may be considered as a clear
indication that the spin-spin repulsion force might actually be by far stronger
than expected earlier, when only the balance between two superextreme Kerr
sources was thought possible. We also present the explicit analytical formulas
relating the equilibrium states in the double-Kerr and
double-Reissner-Nordstr\"om configurations.
