We present a method for sketch-based modelling of 3D man-made shapes that exploits not only the commonly considered visible surface lines but also the hidden lines typical for technical drawings. Hidden lines are used by artists and designers to communicate holistic shape structure. Given a single viewpoint sketch, leveraging such lines allows us to resolve the ambiguity of the shape's surfaces hidden from the observer. We assume that the separation into visible and hidden lines is given, and focus solely on how to leverage this information. Our strategy is to mingle two distinct diffusion networks: one generates denoized occupancy grid estimates from a visible line image, whilst the other generates occupancy grid estimates based on contextualized hidden lines unveiling the occluded shape structure. We iteratively merge noisy estimates from both models in a reverse diffusion process. Importantly, we demonstrate the importance of what we call a contextualized hidden lines image over just a hidden lines image. Our contextualized hidden lines image contains hidden lines and silhouette lines. Such contextualization allows us to achieve superior performance to a range of alternative configurations and reconstruct hidden holes and hidden surfaces.