In 2006, Böhm and Wilking introduced a technique to construct new Ricci flow invariant curvature conditions from known ones, in order to verify a conjecture by Hamilton, that in all dimensions, a compact manifold with positive curvature operator is a spherical space form. We apply this technique to a closed convex curvature condition, which is invariant under the ordinary equation $\frac{d \mathrm{Rm}}{dt} = \mathrm{Rm}^{2} + \mathrm{Rm}^{\#}$ and contains the cone of nonnegative curvature operators. We show that under a certain rescaling and flipping of the traceless Ricci part, one obtains a new curvature condition with the same properties. Via Hamilton's maximum principle, this defines a Ricci flow invariant curvature condition, which in this sense is dual to the original curvature condition. A criterion by Hoelzel yields stability under surgeries of the dual curvature condition, meaning that there exist lots of examples satisfying this condition. Moreover, using similar ideas as Böhm and Wilking, we construct a continous family of cones, which are invariant under the above ordinary equation, joining the dual cone and the invariant cone of curvature operators with nonnegative isotropic curvature.