Aims. Although from a theoretical point of view magnetic fields are believed to play a significant role during the early stages of star formation, especially during the main accretion phase, the magnetic field strength and topology is poorly constrained in the youngest accreting Class 0 protostars that lead to the formation of solar-type stars. Methods. We carried out observations of the polarized dust continuum emission with the SMA interferometer at 0.87 mm to probe the structure of the magnetic field in a sample of 12 low-mass Class 0 envelopes in nearby clouds, including both single protostars and multiple systems. Our SMA observations probed the envelope emission at scales ~600 − 5000 au with a spatial resolution ranging from 600 to 1500 au depending on the source distance. Results. We report the detection of linearly polarized dust continuum emission in all of our targets with average polarization fractions ranging from 2% to 10% in these protostellar envelopes. The polarization fraction decreases with the continuum flux density, which translates into a decrease with the H₂ column density within an individual envelope. Our analysis show that the envelope-scale magnetic field is preferentially observed either aligned or perpendicular to the outflow direction. Interestingly, our results suggest for the first time a relation between the orientation of the magnetic field and the rotational energy of envelopes, with a larger occurrence of misalignment in sources in which strong rotational motions are detected at hundreds to thousands of au scales. We also show that the best agreement between the magnetic field and outflow orientation is found in sources showing no small-scale multiplicity and no large disks at ~100 au scales.