  I examine two cosmological scenarios in which the thermodynamic arrow of time
points in opposite directions in the asymptotic past and future. The first
scenario, suggested by Aguirre and Gratton, assumes that the two asymptotic
regions are separated by a de Sitter-like bounce, with low-entropy boundary
conditions imposed at the bounce. Such boundary conditions naturally arise from
quantum cosmology with Hartle-Hawking wave function of the universe. The bounce
hypersurface breaks de Sitter invariance and represents the beginning of the
universe in this model. The second scenario, proposed by Carroll and Chen,
assumes some generic initial conditions on an infinite spacelike Cauchy
surface. They argue that the resulting spacetime will be non-singular, apart
from black holes that could be formed as the initial data is evolved, and will
exhibit eternal inflation in both time directions. Here I show, assuming the
null convergence condition, that the Cauchy surface in a non-singular (apart
from black holes) universe with two asymptotically inflating regions must
necessarily be compact. I also argue that the size of the universe at the
bounce between the two asymptotic regions cannot much exceed the de Sitter
horizon. The spacetime structure is then very similar to that in the
Aguirre-Gratton scenario and does require special boundary conditions at the
bounce. If cosmological singularities are allowed, then an infinite Cauchy
surface with `random' initial data will generally produce inflating regions in
both time directions. These regions, however, will be surrounded by
singularities and will have singularities in their past or future.
