Channel reciprocity greatly facilitates downlink precoding in time-division duplexing (TDD) multiple-input multiple-output (MIMO) communications without the need for channel state information (CSI) feedback. Recently, reconfigurable intelligent surfaces (RISs) emerge as a promising technology to enhance the performance of future wireless networks. However, since the artificial electromagnetic characteristics of RISs do not strictly follow the normal laws of nature, it brings up a question: does the channel reciprocity hold in RIS-assisted TDD wireless networks? After briefly reviewing the reciprocity theorem, in this article, we show that there still exists channel reciprocity for RIS-assisted wireless networks satisfying certain conditions. We also experimentally demonstrate the reciprocity at the sub-6 GHz and the millimeter-wave frequency bands by using two fabricated RISs. Furthermore, we introduce several RIS-assisted approaches to realizing nonreciprocal channels. Finally, potential opportunities brought by reciprocal/nonreciprocal RISs and future research directions are outlined.