A novel approach to the design of a semantic, low-dimensional,encoding for endoscopic imagery is proposed. This encoding is based on recent advances in scene recognition, where semantic modeling of image content has gained considerable attention over the last decade. While the semantics of scenes are mainly comprised of environmental concepts such as vegetation, mountains or sky, the semantics of endoscopic imagery are medically relevant visual elements, such as polyps, special surface patterns, or vascular structures. The proposed semantic Encoding differs from the representations commonly used in endoscopic image analysis (for medical decision support) in that it establishes a semantic space, where each coordinate axis has a clear human interpretation. It is also shown to establish a connection to Riemannian geometry, which enables principled solutions to a number of problems that arise in both physician training and clinical practice. This connection is exploited by leveraging results from information geometry to solve problems such as 1) recognition of important semantic concepts, 2) semantically-focused image browsing, and 3) estimation of the average-case semantic encoding for a collection of images that share a medically relevant visual detail. The approach can provide physicians with an easily interpretable, semantic encoding of visual content, upon which further decisions, or operations, can be naturally carried out. This is contrary to the prevalent practice in endoscopic image analysis for medical decision support, where image content is primarily captured by discriminative, high-dimensional, appearance features, which possess discriminative power but lack human interpretability.