It is shown that the Delta J=4 bifurcation effect recently discovered
in the superdeformed bands of heavy nuclei also occurs in the rotation
al bands of diatomic molecules in the form of a Delta J=2 staggering.
A possible explanation for this phenomenon assumes a perturbation betw
een rotational levels that can be described in terms of parity-split q
-deformed rotators interacting through a q-deformed, J-dependent inter
action. The phenomenon is evidenced and the model is illustrated on th
e rotational bands of AgH.