We present reconstruction methods for radial magnetic resonance imaging (MRI) data which were spatially encoded using a pair of orthogonal multipolar magnetic fields for in-plane encoding and parallel imaging. It is shown that a direct method exists in addition to iterative reconstruction. Standard direct projection reconstruction algorithms can be combined with a previously developed direct reconstruction for multipolar encoding fields acquired with Cartesian trajectories. The algorithm is simplified by recasting the reconstruction problem into polar coordinates. In this formulation distortion and aliasing become separate effects. Distortion occurs only along the radial direction and aliasing along the azimuthal direction. Moreover, aliased points are equidistantly distributed in this representation, and, consequently, Cartesian SENSE is directly applicable with highly effective unfolding properties for radio-frequency coils arranged with a radial symmetry. The direct and iterative methods are applied to simulated data to analyze basic properties of the algorithms and for the first time also measured in vivo data are presented. The results are compared to linear spatial encoding using a radial trajectory and quadrupolar encoding using a Cartesian trajectory. The direct reconstruction gives good results for fully sampled datasets. Undersampled datasets, however, show star-shaped artifacts, which are significantly reduced with the iterative reconstruction.