  We study a one-parameter family of time-reversible Hamiltonian vector fields
in $\mathbb{R}^4$, which has received great attention in the literature. On the
one hand, it is due to the role it plays in the context of certain applications
in the field of Physics or Engineering and, on the other hand, we especially
highlight its relevance within the framework of generic unfoldings of the
four-dimensional nilpotent singularity of codimension four.
  The system exhibits a bifocal equilibrium point for a range of parameter
values. The associated two-dimensional invariant manifolds, stable and
unstable, fold into the phase space in such a way that they produce intricate
patterns. This entangled geometry has previously been called \textit{tentacular
geometry}. We consider a three-dimensional level set containing the bifocal
equilibrium point to gain insight into the folding behavior of these invariant
manifolds. Our method consists of describing the traces left by invariant
manifolds when crossing an invariant cross section by the reversibility map.
With this new approach, we provide a better understanding of how tentacular
geometry evolves with respect to the parameter.
  Our techniques enables us to link the tentacular geometry on the cross
section with the study of cocooning cascades of homoclinic tangencies. Indeed,
we present a general theory to extend to $\mathbb{R}^4$ the phenomena related
to cocoon bifurcation that classically develop within families of
three-dimensional reversible vector fields. On the basis of these results, we
conjecture the existence of heteroclinic cycles consisting of two orbits
connecting the bifocus with a saddle node periodic orbit.
