  This paper presents a mechanism for the coexistence of hyperbolic and
non-hyperbolic dynamics arising in a neighbourhood of a conservative Bykov
cycle where trajectories turn in opposite directions near the two saddle-foci.
We show that {within the class of divergence-free vector fields that preserve
the cycle,} tangencies of the invariant manifolds of two hyperbolic saddle-foci
densely occur. The global dynamics is persistently dominated by heteroclinic
tangencies and by the existence of infinitely many elliptic points coexisting
with suspended hyperbolic horseshoes. A generalized version of the Cocoon
bifurcations for conservative systems is obtained.
