We report novel spectropolarimetric observations of sunspots carried out with the Tenerife Infrared Polarimeter in a near-IR spectral region around 15410 Å, which is known to contain two groups of prominent OH lines that show circular polarization signals of opposite polarity. Surrounding these well-known OH lines, we have discovered the presence of CN lines of the Δv=1 band that show anomalous polarization profiles. Although the Stokes V signals of the OH lines are antisymmetric and of a sizable amplitude, the CN lines show almost negligible circular polarization. On the contrary, the linear polarization signals turn out to be much stronger in the CN lines than in the OH lines. Interestingly, these CN lines present striking antisymmetric linear polarization profiles, which we are able to explain and model via the Paschen-Back effect theory for diatomic molecules. The presence of such peculiar CN lines in the same spectral region of the OH lines may be useful to improve our empirical knowledge of solar magnetic fields via the simultaneous observation and modeling of the transverse and longitudinal Zeeman effects in two different molecular species.