Mangle deals accurately and efficiently with complex angular masks, such as occur typically in galaxy surveys. Mangle performs the following tasks: converts masks between many handy formats (including HEALPix); rapidly finds the polygons containing a given point on the sphere; rapidly decomposes a set of polygons into disjoint parts; expands masks in spherical harmonics; generates random points with weights given by the mask; and implements computations for correlation function analysis. To mangle, a mask is an arbitrary union of arbitrarily weighted angular regions bounded by arbitrary numbers of edges. The restrictions on the mask are only (1) that each edge must be part of some circle on the sphere (but not necessarily a great circle), and (2) that the weight within each subregion of the mask must be constant. Mangle is complementary to and integrated with the HEALPix package (ascl:1107.018); mangle works with vector graphics whereas HEALPix works with pixels.