Doppler velocities in the penumbra of a δ-configuration sunspot observed near the limb indicate flows that converge upon the line separating locally positive and negative polarity magnetic field (the polarity inversion line). These flows persist for many hours. Observations of this region with the Advanced Stokes Polarimeter (ASP) reveal a convex vector field geometry with magnetic lines of force arching upward from positive polarity, then downward to negative polarity. The straightforward interpretation of the combined Doppler velocity and vector field information leads to an untenable physical situation: were flows directed from both footpoints toward the tops of arched magnetic lines of force, mass would rapidly load the tops of the arches. However, there is no observational evidence of the dynamics that such a loading would require. To better understand this apparent contradiction, we perform two-component analyses of the observed Stokes spectral profiles in the vicinity of the polarity inversion line, in order to extract information about unresolved structure of the magnetic field and its associated flows. Fits to the observed profiles, obtained by use of two different inversion techniques, suggest strongly that, as in penumbrae of simple sunspots, the field geometry in the convergence zone is ``fluted.'' However, unlike in simple sunspots, which have only an outward-directed Evershed flow in the more horizontal of the field components, at each spatial point our analysis reveals flows in the two components that are oppositely directed. We interpret these observations as indicative of an interleaved system of field lines in the vicinity of the polarity reversal, whereby the convergent streams are able to slip past one another and return downward into the solar interior.