One hundred and twenty-five new high-precision spectropolarimetric observations have been obtained with ESPaDOnS (Eschelle Spectro-Polarimetric Device for the Observation of Stars) at the Canada–France–Hawaii Telescope and Narval at Télescope Bernard Lyot to investigate the magnetic properties of the classical Be star ω Ori. No Stokes V signatures are detected in our polarimetric data. Measurements of the longitudinal magnetic field, with a median error bar of 30 G, and direct modelling of the mean least-squares deconvolved Stokes V profiles yield no evidence for a dipole magnetic field with polar surface strength greater than ∼80 G. We are therefore unable to confirm the presence of the magnetic field previously reported by Neiner et al. However, our spectroscopic data reveal the presence of periodic emission variability in H and He lines analogous to that reported by Neiner et al., considered as evidence of magnetically confined circumstellar plasma clouds. We revisit this hypothesis in light of the new magnetic analysis. Calculation of the magnetospheric Kepler radius RK and confinement parameter η* indicates that a surface dipole magnetic field with a polar strength larger than 63 G is sufficient to form of a centrifugally supported magnetosphere around ω Ori. Our data are not sufficiently sensitive to detect fields of this magnitude; we are therefore unable to confirm or falsify the magnetic cloud hypothesis. Based on our results, we examine three possible scenarios that could potentially explain the behaviour of ω Ori: (1) that no significant magnetic field is (or was) present in ω Ori, and that the observed phenomena have their origin in another mechanism or mechanisms than corotating clouds. We are, however, unable to identify one; (2) that ω Ori hosts an intermittent magnetic field produced by dynamo processes; however, no such process has been found so far to work in massive stars and especially to produce a dipolar field; and (3) that ω Ori hosts a stable, organized (fossil) magnetic field that is responsible for the observed phenomena, but with a strength that is below our current detection threshold. Of these three scenarios, we consider the second one (dynamo process) as highly unlikely, whereas the other two should be falsifiable with intense monitoring.