On some invariants of orbits in the flag variety under a symmetric subgroup

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\mathbf{k}$ of characteristic not equal to 2, let $\text{B}$ be the variety of all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing a closed $K$-orbit in $\text{B}$, we associate to every $K$-orbit on $\text{B}$ some subsets of the Weyl group of $G$, and we study them as invariants of the $K$-orbits. When $\mathbf{k}=\mathbb{C}$, these invariants are used to determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup of $G$ have non-empty intersection in $\text{B}$. We also characterize the invariants in terms of admissible paths in the set of $K$-orbits in $\text{B}$.