On intersections of certain partitions of a group compactification

Let $G$ be a connected semi-simple algebraic group of adjoint type over an algebraically closed field, and let $\overline{G}$ be the wonderful compactification of $G$. For a fixed pair $(B, B^-)$ of opposite Borel subgroups of $G$, we look at intersections of Lusztig's $G$-stable pieces and the $B^-\times B$-orbits in $\overline{G}$, as well as intersections of $B \times B$-orbits and $B^- \times B^-$-orbits in $\overline{G}$. We give explicit conditions for such intersections to be non-empty, and in each case, we show that every non-empty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the non-empty intersections form a strongly admissible partition of $\overline{G}$.