Solar magnetic activity varies with time in the two hemispheres in different ways. The hemispheric interconnection of solar activity phenomena provides an important clue to understanding the dynamical behavior of solar dynamo actions. In this paper, several analysis approaches are proposed to analyze the systematic regularity of hemispheric asynchronism and amplitude asymmetry of long-term sunspot areas during solar cycles 9–24. It is found that, (1) both the hemispheric asynchronism and the amplitude asymmetry of sunspot areas are prevalent behaviors and are not anomalous, but the hemispheric asynchronism exhibits a much more regular behavior than the amplitude asymmetry; (2) the phase-leading hemisphere returns back to the identical hemisphere every 8 solar cycles, and the secular periodic pattern of hemispheric phase differences follows 3 (south leading) + 5 (north leading) solar cycles, which probably corresponds to the Gleissberg cycle; and (3) the pronounced periodicities of (absolute and normalized) asymmetry indices and lines of synchronization (LOSs) are not identical: the significant periodic oscillations are 80.65 ± 6.31, 20.91 ± 0.40, and 13.45 ± 0.16 years for the LOS values, and 51.34 ± 2.48, 8.83/8.69 ± 0.07, and 3.77 ± 0.02 years for the (absolute and normalized) asymmetry indices. The analysis results improve our knowledge on the hemispheric interrelation of solar magnetic activity and may provide valuable constraints for solar dynamo models.