For a connected reductive group G and a Borel subgroup B, we study the
closures of double classes BBB in a (G x G)-equivariant ''regular'' c
ompactification of G. We show that these closures (BgB) over bar inter
sect properly all (G x G)-orbits, with multiplicity one, and we descri
be the intersections. Moreover, we show that almost all (BgB) over bar
are singular in codimension two exactly. We deduce this from more gen
eral results on B-orbits in a spherical homogeneous space G/H; they le
ad to formulas for homology classes of X-orbit closures in G/B, in ter
ms of Schubert cycles.