  We investigate a model in which spinors are considered as being embedded
within the Clifford algebra that operates on them. In Minkowski space
$M_{1,3}$, we have four independent 4-component spinors, each living in a
different minimal left ideal of $Cl(1,3)$. We show that under space inversion,
a spinor of one left ideal transforms into a spinor of another left ideal. This
brings novel insight to the role of chirality in weak interactions. We
demonstrate the latter role by considering an action for a generalized spinor
field $\psi^{\alpha i}$ that has not only a spinor index $\alpha$ but also an
extra index $i$ running over four ideals. The covariant derivative of
$\psi^{\alpha i}$ contains the generalized spin connection, the extra
components of which are interpreted as the SU(2) gauge fields of weak
interactions and their generalization. We thus arrive at a system that is
left-right symmetric due to the presence of a "parallel sector", postulated a
long time ago, that contains mirror particles coupled to mirror SU(2) gauge
fields.
