The linear polarization produced by scattering processes in the hydrogen Lyα line of the solar disk radiation is a key observable for probing the chromosphere–corona transition region (TR) and the underlying chromospheric plasma. While the line-center signal encodes information on the magnetic field and the three-dimensional structure of the TR, the sizable scattering polarization signals that the joint action of partial frequency redistribution and J-state interference produce in the Lyα wings have generally been thought to be sensitive only to the thermal structure of the solar atmosphere. Here we show that the wings of the Q/I and U/I scattering polarization profiles of this line are actually sensitive to the presence of chromospheric magnetic fields, with strengths similar to those that produce the Hanle effect in the line core (i.e., between 5 and 100 G, approximately). In spite of the fact that the Zeeman splitting induced by such weak fields is very small compared to the total width of the line, the magneto-optical effects that couple the transfer equations for Stokes Q and U are actually able to produce sizable changes in the Q/I and U/I wings. We find that magnetic fields with longitudinal components larger than 100 G produce an almost complete depolarization of the wings of the Lyα Q/I profiles within a ±5 Å spectral range around the line center, while stronger fields are required for the U/I wing signals to be depolarized to a similar extent. The theoretical results presented here further expand the diagnostic content of the unprecedented spectropolarimetric observations provided by the Chromospheric Lyman-Alpha Spectro-Polarimeter.