Let $G$ be a classical group defined over the complex numbers with a Borel subgroup $B$. Choose a holomorphic involution of $G$ and let $K$ be its set of fixed points. The group $K$ acts on the flag variety $G/B$ with finitely many orbits and Brion has derived a general formula for the cohomology classes of the corresponding orbit closures as linear combinations of Schubert classes. This article provide a uniform description of the sets of Weyl group elements (which we refer to as Brion atoms) indexing the terms in this formula. This builds on prior work addressing types A, B, and C. The main novelty of our results is a thorough treatment of type D. As one application, we introduce a notion of involution Schubert polynomials for all classical types and present several conjectures related to these objects.