Dual-energy computed tomography (DECT) plays a crucial role in clinical practice due to its capabilities in material identification and quantification. However, complete DECT scans expose patients to additional radiation. To mitigate such a problem, one method is to reduce the CT scan angular range, which may cause limited-angle (LA) artifacts. This work utilized multistage dual-domain networks (MsDu-Nets) for dual-energy LA reconstruction that only requires a 180° scan. MsDu-Nets consist of multiple stages, where data processing is performed in both the sinogram domain and image domain within each stage. A filtered back projection (FBP) module is used to connect these two domains. MsDu-Nets leveraged the informative prior obtained by directly concatenating the dual-energy projection data and directly reconstructing it after applying Parker weights, which helps the network focus more on anatomical information. Self-augmentation is also employed to mitigate the streak artifacts introduced by the informative prior and enables the transmission of original information across stages. Comparison experiments and material decomposition experiments demonstrate that the proposed method is capable of obtaining high-quality dual-energy reconstructed images. The ablation experiments also validate the roles of different modules or components of the method.