Using the rotation rates of the solar magnetic field during solar cycles 21 to 23 obtained by Chu et al. by analyzing the synoptic magnetic maps produced by the NSO/Kitt Peak and SOHO/MDI during the years 1975 to 2008, the temporal variation of the equatorial rotation rate (A) and the latitude gradient of rotation (B) in the northern and southern hemispheres are studied separately. The results indicate that the rotation is more differential (about 4.3%) in the southern hemisphere in the considered time frame. It is found that the north-south asymmetry of A and the asymmetry of B show increasing trends in the considered time frame, while the north-south asymmetry of the solar activity shows a decreasing trend. There exists a significant negative correlation (at 95% confidence level) between the asymmetry of B and the asymmetry of the solar activity, and this may be due to stronger magnetic activity in a certain hemisphere that may suppress the differential rotation to some extent. The periodicities in the variation of A and B are also studied, and periods of about 5.0 and 10.5 yr (5.5 and 10.4 yr) can be found for the variation of the northern (southern) hemisphere B. Moreover, the north-south asymmetry of A and the asymmetry of B have similar periods of about 2.6-2.7 and 5.2-5.3 yr. Further, cross-correlation analysis indicates that there exists a phase difference (about eight months) between the northern and southern hemisphere B, and this means that the northern hemisphere B generally leads by about eight months.