If we are asked to visualize a Möbius band, we do not think first of its symmetry. However, if we make a model of a Möbius band with a computer program (for example, with Maple) and examine its boundary from different points of view, we get various interesting, symmetrical figures. A model of a Möbius band can be constructed by joining the ends of a strip (long rectangle) of paper with a single half-twist. It is interesting to observe how the resulting band transforms as we vary the ratio between the long and short sides of the rectangle. When will the surface intersect itself? We shall analyse these problems with multiply-twisted strips. The second part of this article deals with the connection between the Möbius band and frieze patterns.