  (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.
