X-ray images are widely used in the study of paintings. When a painting has hidden sub-surface features (e.g., reuse of the canvas or revision of a composition by the artist), the resulting X-ray images can be hard to interpret as they include contributions from both the surface painting and the hidden design. In this paper we propose a self-supervised deep learning-based image separation approach that can be applied to the X-ray images from such paintings ('mixed X-ray images') to separate them into two hypothetical X-ray images, one containing information related to the visible painting only and the other containing the hidden features. The proposed approach involves two steps: (1) separation of the mixed X-ray image into two images, guided by the combined use of a reconstruction and an exclusion loss; (2) even allocation of the error map into the two individual, separated X-ray images, yielding separation results that have an appearance that is more familiar in relation to X-ray images. The proposed method was demonstrated on a real painting with hidden content, Doña Isabel de Porcel by Francisco de Goya, to show its effectiveness.