  (abridged) As the favoured progenitors of long-duration gamma-ray bursts, massive stars may represent our best signposts of individual objects in the early Universe, but special conditions seem required to make these bursters, which might originate from the progenitor's rapid rotation and associated asymmetry. To obtain empirical constraints on the interplay between stellar rotation and wind asymmetry, we perform linear Halpha spectropolarimetry on a sample of 18 spectroscopically peculiar massive O stars, including OVz, Of?p, Oe, and Onfp stars, supplemented by an earlier sample of 20 O supergiants. Despite their rapid rotation (with vsin(i) up to 400 km/s) most O-type stars are found to be spherically symmetric, but with notable exceptions amongst specific object classes. We divide the peculiar O stars into four distinct categories: Groups III and IV include the Oe stars and Onfp stars, which are on the high-end tail of the O star rotation distribution and have in the past been claimed to be embedded in disks. Here we report the detection of a classical depolarization ``line effect'' in the Oe star HD 45314, but the overall incidence of line effects amongst Oe stars is significantly lower (1 out of 6) than amongst Be stars. The chance that the Oe and Be datasets are drawn from the same parent population is negligible (with 95% confidence). This implies there is as yet no evidence for a disk hypothesis in Oe stars, providing relevant constraints on the physical mechanism that is responsible for the Be phenomenon. Finally, we find that 3 out of 4 of the group IV Onfp stars show evidence for complex polarization effects, which are likely related to rapid rotation, and we speculate on the evolutionary links to B[e] stars. 
On the presence and absence of disks around O-type stars