Solar magnetic fields leave their fingerprints in the polarization signatures of the emergent spectral line radiation. This occurs through a variety of rather unfamiliar physical mechanisms, not only via the Zeeman effect. In particular, magnetic fields modify the atomic level polarization (population imbalances and quantum coherences) that anisotropic radiative pumping processes induce in the atoms and molecules of the solar atmosphere. Interestingly, this so-called Hanle effect allows us to ``see'' magnetic fields to which the Zeeman effect is blind within the limitations of the available instrumentation. Here I argue that the Ca II IR triplet and the He I 10830 Å multiplet would be very suitable choices for investigating the magnetism of the solar chromosphere via spectropolarimetric observations from a future space telescope, such as JAXA's SOLAR-C mission.