The need for comprehensive telemedicine solutions is becoming increasingly relevant due to challenges associated with the ageing population, the increasing shortage of health-care providers, and, more recently, the global pandemic. Existing solutions primarily focus on, e.g., electronic medical records, audiovisual connections, and, in some cases, robotic systems with very basic capabilities. Here we present a fundamentally new, holistic approach to a remote doctor visit, which enables transparent remote examination, anomaly detection, diagnosis, and rehabilitation. Our dual doctor-patient twin paradigm involves two robotic systems: one representing the doctor to the patient ("GARMI") and one representing the patient to the doctor ("MUCKI"). Through bidirectional telepresence control, this system enables transparent, natural, remote haptic interaction between doctor and patient. The control, interaction, and knowledge transfer to the doctor is enhanced by AI-based visual motion and facial expression analysis as well as a digital twin of the patient. Thus, each stage of a doctor visit can be replicated in the context of telemedicine and shared autonomy: from first assessment to observation-based and remote physical examination, to a better-informed doctor diagnosis and robot-assisted telerehabilitation.