Camouflaged object detection, which aims to detect/segment the object(s) that blend in with their surrounding, remains challenging for deep models due to the intrinsic similarities between foreground objects and background surroundings. Ideally, an effective model should be capable of finding valuable clues from the given scene and integrating them into a joint learning framework to co-enhance the representation. Inspired by this observation, we propose a novel Mutual Graph Learning (MGL) model by shifting the conventional perspective of mutual learning from regular grids to graph domain. Specifically, an image is decoupled by MGL into two task-specific feature maps - one for finding the rough location of the target and the other for capturing its accurate boundary details. Then, the mutual benefits can be fully exploited by reasoning their high-order relations through graphs recurrently. It should be noted that our method is different from most mutual learning models that model all between-task interactions with the use of a shared function. To increase information interactions, MGL is built with typed functions for dealing with different complementary relations. To overcome the accuracy loss caused by interpolation to higher resolution and the computational redundancy resulting from recurrent learning, the S-MGL is equipped with a multi-source attention contextual recovery module, called R-MGL_v2, which uses the pixel feature information iteratively. Experiments on challenging datasets, including CHAMELEON, CAMO, COD10K, and NC4K demonstrate the effectiveness of our MGL with superior performance to existing state-of-the-art methods. The code can be found at https://github.com/fanyang587/MGL.