I'll be using the diagrams here to explain a bit:
In dichroic atomic vapor spectroscopy you have a constant longitudinal field through a vapor cell, you send linearly polarized light through the cell, and then measure the output on a photodetector. Because of the Zeeman effect, the energy levels split and you can find different transition frequencies through spectroscopy. So if you probe the cell with linearly polarized light with frequency $f_0$ (say, corresponding to the D1-line of rubidium), then you actually see two red-shifted and blue-shifted absorption peaks; this is because the linearly polarized light can be seen as a superposition of left-handed and right-handed circularly polarized light.
I'm confused because it doesn't seem (or at least, no one has thought it useful to mention) that it's possible to still see the broad central peak at $f_0$ due to linearly polarized light still interacting with the atoms as linearly polarized. Those transitions are still available in terms of frequency differences (since the Zeeman splitting doesn't offset the $m_l = 0$ line), and I don't believe there are selection rules that I'm overlooking, but it would be a convenient explanation if that were the case.