  Studies of dense core morphologies and their orientations with respect to gas
flows and the local magnetic field have been limited to only a small sample of
cores with spectroscopic data. Leveraging the Green Bank Ammonia Survey
alongside existing sub-millimeter continuum observations and Planck dust
polarization, we produce a cross-matched catalogue of 399 dense cores with
estimates of core morphology, size, mass, specific angular momentum, and
magnetic field orientation. Of the 399 cores, 329 exhibit 2D
$\mathrm{v}_\mathrm{LSR}$ maps that are well fit with a linear gradient,
consistent with rotation projected on the sky. We find a best-fit specific
angular momentum and core size relationship of $J/M \propto R^{1.82 \pm 0.10}$,
suggesting that core velocity gradients originate from a combination of solid
body rotation and turbulent motions. Most cores have no preferred orientation
between the axis of core elongation, velocity gradient direction, and the
ambient magnetic field orientation, favouring a triaxial and weakly magnetized
origin. We find, however, strong evidence for a preferred anti-alignment
between the core elongation axis and magnetic field for protostellar cores,
revealing a change in orientation from starless and prestellar populations that
may result from gravitational contraction in a magnetically-regulated (but not
dominant) environment. We also find marginal evidence for anti-alignment
between the core velocity gradient and magnetic field orientation in the L1228
and L1251 regions of Cepheus, suggesting a preferred orientation with respect
to magnetic fields may be more prevalent in regions with locally ordered
fields.
