  We investigate the cosmological properties of an "induced gravity" brane
scenario in the absence of mirror symmetry with respect to the brane. We find
that brane evolution can proceed along one of four distinct branches. By
contrast, when mirror symmetry is imposed, only two branches exist, one of
which represents the self-accelerating brane, while the other is the so-called
normal branch. This model incorporates many of the well-known possibilities of
brane cosmology including phantom acceleration (w < -1), self-acceleration,
transient acceleration, quiescent singularities, and cosmic mimicry.
Significantly, the absence of mirror symmetry also provides an interesting way
of inducing a sufficiently small cosmological constant on the brane. A small
(positive) Lambda-term in this case is induced by a small asymmetry in the
values of bulk fundamental constants on the two sides of the brane.
