We study the intersection of a positively sectional-hyperbolic set and a negatively sectional-hyperbolic setof a flow on a compact manifold.Indeed, we show that such an intersection is not a hyperbolic set in general.Next, we show that such an intersection is a hyperbolic set if the sets involved in the intersectionare both transitive.In general, we prove that such an intersection is the disjoint union of a nonsingular hyperbolic set, a finite set of singularities,and a set of regular orbits joining these dynamical objects.Finally, we exhibit a three-dimensional star flowwith a positively (but not negatively) sectional-hyperbolic homoclinic class anda negatively (but not positively) sectional-hyperbolic homoclinic class whose intersection is a periodic orbit.This provides a counterexample to a conjecture of Shi, Zhu, Gan and Wen ([25], [26]).