  We make the hypothesis that molecular cloud fragments are triaxial bodies
with a large scale magnetic field oriented along the short axis. While
consistent with theoretical expectations, this idea is supported by magnetic
field strength data, which show strong evidence for flattening along the
direction of the mean magnetic field. It is also consistent with early
submillimeter polarization data, which show that the projected direction of the
magnetic field is often slightly misaligned with the projected minor axis of a
molecular cloud core, i.e., the offset angle $\Psi$ is nonzero. We calculate
distributions of $\Psi$ for various triaxial bodies, when viewed from a random
set of viewing angles. The highest viewing probability always corresponds to
$\Psi=0^{\circ}$, but there is a finite probability of viewing all nonzero
$\Psi$, including even $\Psi =90^{\circ}$; the average offset typically falls
in the range $10^{\circ}-30^{\circ}$ for triaxial bodies most likely to satisfy
observational and theoretical constraints.
