  A closed orientable manifold is {\em achiral} if it admits an orientation
reversing homeomorphism. A commensurable class of closed manifolds is achiral
if it contains an achiral element, or equivalently, each manifold in $\CM$ has
an achiral finite cover.
  Each commensurable class containing non-orientable elements must be achiral.
  It is natural to wonder how many
  commensurable classes are achiral and how many achiral classes have
non-orientable elements.
  We study this problem for Sol 3-manifolds. Each commensurable class $\CM$ of
Sol 3-manifold has a complete topological invariant $D_{\CM}$, the discriminant
of $\CM$. Our main result is:
  (1) Among all commensurable classes of Sol 3-manifolds, there are infinitely
many achiral classes; however ordered by discriminants, the density of achiral
commensurable classes is 0.
  (2) Among all achiral commensurable classes of Sol 3-manifolds, ordered by
discriminants, the density of classes containing non-orientable elements is
$1-\rho$,
  where $$\rho:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1} =
0.41942\cdots.$$
