Using doors is a longstanding challenge in robotics and is of significant practical interest in giving robots greater access to human-centric spaces. The task is challenging due to the need for online adaptation to varying door properties and precise control in manipulating the door panel and navigating through the confined doorway. To address this, we propose a learning-based controller for a legged manipulator to open and traverse through doors. The controller is trained using a teacher-student approach in simulation to learn robust task behaviors as well as estimate crucial door properties during the interaction. Unlike previous works, our approach is a single control policy that can handle both push and pull doors through learned behaviour which infers the opening direction during deployment without prior knowledge. The policy was deployed on the ANYmal legged robot with an arm and achieved a success rate of 95.0% in repeated trials conducted in an experimental setting. Additional experiments validate the policy's effectiveness and robustness to various doors and disturbances. A video overview of the method and experiments can be found at youtu.be/tQDZXN_k5NU.