The errors of inertial sensors affect the navigation accuracy of the strapdown inertial navigation system (SINS) and are accumulated over time in nature. In order to continuously maintain the high navigation accuracy of vehicles for a long time period, an initial alignment and self-calibration is necessary after the SINS starts. Additionally, the observability analysis is one of the key techniques during the initial alignment and self-calibration process. For marine systems, the observability of inertial sensor errors is extremely low, as their motion states are always slow. Therefore, studying the rotating SINS is urgent. Since traditional analysis methods have their limitations, the global observation analysis method was used in this paper. On the basis of this method, the relationship between the observability and the kinestate of the rotating SINS has been established. After the discussion about the factors that affect the observability in detail, the design principle of the initial alignment and self-calibration rotating scheme, which is appropriate for marine systems, id proposed. With the proposed principle, a novel initial alignment and self-calibration method, named the eight-position rotating scheme, is designed. Simulations and experiments are carried out to verify its performance. The results have shown that compared with other rotating schemes and the static state, the estimated accuracy of the eight-position scheme rotating about axes x and y was the best, and the position error was significantly reduced with this new rotating scheme. The feasibility and effectiveness of the proposed design principle and the rotating scheme were verified.