In an old paper, Ehrenfest 1931, the introduction starts off as follows:
The band spectra of symmetric diatomic molecules show certain striking differences from those of asymmetric molecules. For when the two nuclei of the molecule are identical, the intensity of the individual lines of a band, instead of varying smoothly from line to line, alternates more or less markedly.
There is something called the relative weight, defined by the ratio of the number of states that are symmetric under interchange of the two nuclei and the number that are asymmetric. In modern language, let the degeneracy of the nuclear ground state be $g$. Then the relative weight is $(g+1)/(g-1)$ if the nuclei are bosons, $(g-1)/(g+1)$ if fermions. If I'm understanding correctly, then if the rotational state of the molecule is $J$, then depending on whether $J$ is odd or even, the two nuclei can pick up a relative phase $(-1)^J$, so the degeneracy of the odd-$J$ and even-$J$ states alternates according to the relative weights.
Assuming the above is right, then there's only one thing I really don't understand. How does this show up experimentally as an alternating pattern of intensities? What kind of transitions are we talking about? M1? E2? Are these absorption spectra? Emission spectra? Suppose you populate a molecular state with some spin. Then we're just hopping down a ladder, and it seems to me that by simple conservation of probability there can't be any alternation of intensity. Or by intensity do they actually mean transition rate rather than what an experimentalist would call intensity? (If so, then I'm curious how one would determine these transition rates. From natural line widths? From competition between radiation and collisions in a sample of gas?)
P. Ehrenfest and J. R. Oppenheimer, "Note on the Statistics of Nuclei," Phys. Rev. 37 (1931) 333, http://link.aps.org/doi/10.1103/PhysRev.37.333 , DOI: 10.1103/PhysRev.37.333