  We revisit the secular 3D planetary three-body problem aiming to provide a
unified formalism for studying the structure of the phase space for
progressively higher values of the mutual inclination $i_{mut}$ between the two
planets' orbits. We propose a `book-keeping' technique yielding (after Jacobi
reduction) a clear decomposition of the secular Hamiltonian as
$H_{sec}=H_{planar} +H_{space}$, where $H_{space}$ contains all terms depending
on $i_{mut}$. We numerically compare several models obtained via expansion in
the orbital eccentricities or via multipole expansion. We find the mimimum
required truncation orders to accurately represent the dynamics. We explore the
transition, as $i_{mut}$ increases, from a `planar-like' to a `Lidov-Kozai'
regime. Using a typical (non-hierarchical) example, we show how the structure
of the phase portraits of the integrable secular dynamics of the planar case is
reproduced to a large extent also in the 3D case. We estimate semi-analytically
the level of $i_{mut}$ up to which the dynamics remains nearly-integrable. In
this regime, we propose a normal form method by which the basic periodic orbits
of the nearly-integrable regime (apsidal corotation resonances) can be computed
semi-analytically. On the other hand, as the energy increases the system
gradually moves to the `Lidov-Kozai' regime. The latter is dominated by two
different families of inclined periodic orbits ($C_1$ and $C_2$), of which
$C_2$ becomes unstable via the usual Lidov-Kozai mechanism. We discuss the
connection between the above families of periodic orbits. Finally, we study
numerically the form of the phase portraits for different mass and semi-major
axis ratios of the two planets, aiming to establish how generic are the
phenomena reported above as the systems parameters are chosen close to one or
more hierarchical limits.
