We present 350 μm polarization observations of four low-mass cores containing Class 0 protostars: L483, L1157, L1448-IRS2, and Serp-FIR1. This is the second paper in a larger survey aimed at testing magnetically regulated models for core-collapse. One key prediction of these models is that the mean magnetic field in a core should be aligned with the symmetry axis (minor axis) of the flattened young stellar object inner envelope (aka pseudodisk). Furthermore, the field should exhibit a pinched or hourglass-shaped morphology as gravity drags the field inward toward the central protostar. We combine our results for the four cores with results for three similar cores that were published in the first paper from our survey. An analysis of the 350 μm polarization data for the seven cores yields evidence of a positive correlation between mean field direction and pseudodisk symmetry axis. Our rough estimate for the probability of obtaining by pure chance a correlation as strong as the one we found is about 5%. In addition, we combine together data for multiple cores to create a source-averaged magnetic field map having improved signal-to-noise ratio, and this map shows good agreement between mean field direction and pseudodisk axis (they are within 15°). We also see hints of a magnetic pinch in the source-averaged map. We conclude that core-scale magnetic fields appear to be strong enough to guide gas infall, as predicted by the magnetically regulated models. Finally, we find evidence of a positive correlation between core magnetic field direction and bipolar outflow axis. © 2013. The American Astronomical Society. All rights reserved..