Descent-cycling in Schubert calculus

We prove two lemmata about Schubert calculus on generalized flag manifolds G/B, and in the case of the ordinary flag manifold GL(n)/B we interpret them combinatorially in terms of descents, and geometrically in terms of missing subspaces. One of them gives a symmetry of Schubert calculus that we christen descent-cycling. Computer experiment shows these two lemmata are surprisingly powerful: they already suffice to determine all of GL(n) Schubert calculus through n=5, and 99.97%+ at n=6. We use them to give a quick proof of Monk's rule. The lemmata also hold in equivariant ("double") Schubert calculus for Kac-Moody groups G.