Studying solar rotation, its temporal variation and altitude dependence in the solar atmosphere is timely and important since it is closely related to the mechanism of solar activity. The aim of the present work is to investigate solar differential rotation using rotation rates of Hα filaments in 1957–1993 and those of compact elements of large-scale magnetic fields in 1966–1985. We verify the earlier finding that the differential rotation of both of these magnetic structures is north–south (N–S) asymmetric during the respective time intervals. We find that the N–S asymmetry of Hα filaments changes its sign during the 22-year cycle. The average rotation rate of compact magnetic elements with negative polarity depicts a 22-year oscillation both in the northern and southern hemisphere. Our statistical analysis did not reveal significant N–S asymmetry in the differential rotation of compact elements with positive polarity. Our results on the N–S asymmetry of Hα filaments and compact elements are somewhat controversial since they show opposite simultaneous N–S asymmetries. This may be due to their different heights in the solar atmosphere where radial differential rotation may cause changes in asymmetry.