  There are numerous examples of approximately degenerate states of opposite
parity in molecular physics. Theory indicates that these doubles can occur in
molecules that are reflection-asymmetric. Such parity doubles occur in nuclear
physics as well, among nuclei with odd A $\sim$ 219-229. We have also suggested
elsewhere that such doubles occur in particle physics for baryons made up of
`cbu' and `cbd' quarks. In this article, we discuss the theoretical foundations
of these doubles in detail, demonstrating their emergence as a surprisingly
subtle consequence of the Born-Oppenheimer approximation, and emphasizing their
bundle-theoretic and topological underpinnings. Starting with certain ``low
energy'' effective theories in which classical symmetries like parity and time
reversal are anomalously broken on quantization, we show how these symmetries
can be restored by judicious inclusion of ``high-energy'' degrees of freedom.
This mechanism of restoring the symmetry naturally leads to the aforementioned
doublet structure. A novel by-product of this mechanism is the emergence of an
approximate symmetry (corresponding to the approximate degeneracy of the
doubles) at low energies which is not evident in the full Hamiltonian. We also
discuss the implications of this mechanism for Skyrmion physics, monopoles,
anomalies and quantum gravity.
