That's {absence of fine structure(often) in absorption spectra} interesting.-
Not so. By necessity the source material is excited for emission, but not for absorption in most cases. When the lower level of the transition is not populated, there will be no absorption corresponding to it.
I think that some experiment on absorption in gas that is emitting with fine structure do show fine structure "resonate scattering" which is absorption followed by essentially immediate re-radiation of the beam sent in. I have seen demonstration of laser light passed thru small volume of gas that absorbs it (but probably was not any thing directly related to fine structure) make the entire volume glow with the laser light. It is in this type of situation that the absorption fine spectra can be seen, I am almost certain.
Also, I think it is easy, or would be*, to demonstrate that hydrogen gas will resonantly absorb / re-radiate = scatter the 21 cm line, which corresponds to the nuclear spin flip of the one nucleus in the molecule H2. (The fine structure is related to these nuclear spin orientations, as I recall.) Part of the problem is, I think, also related to need to have highly resolved luminous sources for demonstration that the fine structure is active in absorption.
I am quite rusty on all this, but perhaps some one who is not can comment/contribute along these lines.
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* because of the wavelength, it might take a room full of hydrogen to from a well defined beam of 21cm waves passing thru small part of room - perhaps too dangerous to have been done?