We present Hubble Space Telescope (HST) Wide Field Planetary Camera 2 and deep ground-based images of the Red Rectangle (RR), a bipolar proto - planetary nebula associated with the post-AGB binary system HD 44179. The high-resolution HST images reveal complex new structures, many of them unique to this object. The RR nebula is dominated by a discontinuous "bicone,'' whose bright, sharp linear edges give the nebula an overall X-shaped appearance. The edges of the bicone are connected by a remarkable series of linear features elongated perpendicular to the radius vector, giving the object a ladder-like structure. The "rungs'' of the ladder structure show a quasi-periodic spacing, suggesting that they have arisen from discrete episodes of mass loss from the central star, separated by a few hundred years. The total timescale over which mass has been shed into the visible nebula is of order 14,000 yr. Outside the X-shaped bicone, parabolas curl inward, resembling wineglasses, which terminate on the bicone edges in large, limb-brightened vortices. The central object is bisected by a dark band, indicating that the star is not seen directly but is instead obscured by a surrounding opaque dust disk.