Recent advances in the fabrication and experimentation of Reconfigurable Intelligent Surfaces (RISs) have motivated the concept of the smart radio environment, according to which the propagation of information-bearing waveforms in the wireless medium is amenable to programmability. Although the vast majority of recent experimental research on RIS-empowered wireless communications gravitates around narrowband beamforming in quasi-free space, RISs are foreseen to revolutionize wideband wireless connectivity in dense urban as well as indoor scenarios, which are usually characterized as strongly reverberant environments exhibiting severe multipath conditions. In this article, capitalizing on recent physics-driven experimental explorations of RIS-empowered wave propagation control in complex scattering cavities, we identify the potential of the spatiotemporal control offered by RISs to boost wireless communications in rich scattering channels via two case studies. First, an RIS is deployed to shape the multipath channel impulse response, which is shown to enable higher achievable communication rates. Second, the RIS-tunable propagation environment is leveraged as an analog multiplexer to localize non-cooperative objects using wave fingerprints, even when they are outside the line of sight. Future research challenges and opportunities in the algorithmic design and experimentation of smart rich scattering wireless environments enabled by RISs for sixth Generation (6G) wireless communications are discussed.