The Internet of Things (IoT) applications such as smart grids, smart agriculture, smart cities, and e-healthcare are popular nowadays. Generally, IoT end devices are extremely sensitive to the utilization of energy. The medium access control (MAC) layer is responsible for coordination and access of the IoT devices. It is essential to design an efficient MAC protocol for achieving high throughput in IoT. Duty cycling is a fundamental process in wireless networks and also an energy-saving necessity if nodes are required to operate for more than a few days. Numerous MAC protocols along with different objectives have been proposed for the IoT. However, to the best of our knowledge, only limited work has been performed dedicated to covering MAC and radio duty cycling (RDC). Therefore, in this study, we propose a systematic cataloging system and use if to organize the most important MAC and RDC proposals. In this catalog, each protocol has been categorized into main ideas, advantages, applications, limitations, innovative features, and potential future improvements. Our critical analysis is different from previous research studies, as we have fully covered all recent studies in this domain. We discuss challenges and future research directions.