  We present a rederivation of the baryon and lepton numbers $\frac{1}{2}$ of
the $SU(2)_L$ S sphaleron of the standard electroweak theory based on spectral
mirror symmetry. We explore the properties of a fermionic Hamiltonian under
discrete transformations along a noncontractible loop of field configurations
that passes through the sphaleron and whose endpoints are the vacuum. As is
well known, CP transformation is not a symmetry of the system anywhere on the
loop, except at the endpoints. By augmenting CP with a chirality
transformation, we observe that the Dirac Hamiltonian is odd under the new
transformation precisely at the sphaleron, and this ensures the mirror symmetry
of the spectrum, including the continua. As a consistency check, we show that
the fermionic zero mode presented by Ringwald in the sphaleron background is
invariant under the new transformation. The spectral mirror symmetry which we
establish here, together with the presence of the zero mode, are the two
necessary conditions whence the fermion number $\frac{1}{2}$ of the sphaleron
can be inferred using the reasoning presented by Jackiw and Rebbi or,
equivalently, using the spectral deficiency $\frac{1}{2}$ of the Dirac sea. The
relevance of this analysis to other solutions is also discussed.
