Skinning was initially devised for computing the skin of a character deformed through a skeleton; but it is now also commonly used for deforming tight garments at a very cheap cost. However, unlike skin which may easily compress and stretch, tight cloth strongly resists compression: inside bending regions such as the interior of an elbow, cloth does not shrink but instead buckles, causing interesting folds and wrinkles which are completely missed by skinning methods. Our goal is to extend traditional skinning in order to capture such folding patterns automatically, without sacrificing efficiency. The key of our model is to replace the usual skinning formula — derived from, e.g., Linear Blend Skinning or Dual Quaternions — with a complementarity constraint, making an automatic switch between, on the one hand, classical skinning in zones prone to stretching, and on the other hand, a quasi-isometric scheme in zones prone to compression. Moreover, our method provides some useful handles to the user for directing the type of folds created, such as the fold density or the overall shape of a given fold. Our results show that our method can generate similar complexity of folds compared to full cloth simulation, while retaining interactivity of skinning approaches and offering intuitive user control