  The open intervals in the Bruhat order on twisted involutions in a Coxeter
group are shown to be PL spheres. This implies results conjectured by F.
Incitti and sharpens the known fact that these posets are Gorenstein* over Z_2.
  We also introduce a Boolean cell complex which is an analogue for twisted
involutions of the Coxeter complex. Several classical Coxeter complex
properties are shared by our complex. When the group is finite, it is a
shellable sphere, shelling orders being given by the linear extensions of the
weak order on twisted involutions. Furthermore, the h-polynomial of the complex
coincides with the polynomial counting twisted involutions by descents. In
particular, this gives a type independent proof that the latter is symmetric.
