We prove a root system uniform, concise combinatorial rule for Schubert calculus of minuscule and cominuscule flag manifolds G/P (the latter are also known as compact Hermitian symmetric spaces). We connect this geometry to the poset combinatorics of Proctor, thereby giving a generalization of Schützenberger's jeu de taquin formulation of the Littlewood–Richardson rule that computes the intersection numbers of Grassmannian Schubert varieties. Our proof introduces cominuscule recursions, a general technique to relate the numbers for different Lie types.