The richly structured linearly polarized spectrum that is produced by coherent scattering in the Sun’s atmosphere contains a number of spectral features for which no explanation has been found within the standard scattering theory. According to this quantum- mechanical framework, the intrinsic polarizability of a given line should be determined by the total angular momentum quantum numbers of the atomic levels involved in the scattering transition (which may be resonant or fluorescent). Well defined polarization peaks have been observed in many lines, which according to these theoretical concepts should be intrinsically unpolarizable. A possible explanation for these anomalous spectral structures could be that the initial ground state of the scattering transition becomes polarized by an optical pumping process. However, such an explanation is contradicted by other observations, since it seems to require that much of the solar atmosphere must be filled with extremely weak magnetic fields (~ 10 mG). We have searched through the whole visible solar spectrum for lines with the quantum numbers that should normally make them unpolarizable, and have carried out a systematic observing program for the most prominent of these lines. Here we report on the observed properties of the polarized line profiles of these lines and explain in what respect their behaviors are anomalous and cannot be understood within current conceptual frameworks.