The design principles and special advantages of monolithic adjustable-radius metal mirrors are now well established. Such mirror systems are usually cut from a single block of metal by a wire-electric-discharge machining system, and they consist of a bendable mirror joined to its bending device by thin webs of metal that can be treated as hinges. Analysis is provided for understanding the response of such mirrors to the unintended couples they receive from these flexural hinges when the hinge angle is not zero. The rigidity (the couple per unit angle) of the usual types of flexural hinge and of the most common mirror shapes are calculated thus allowing the hinge-induced distortions of any mirror surface to be estimated. The analysis includes mirrors both with and without water-cooling channels. Two strategies for reducing the errors are proposed. One involves the design of sufficiently flexible hinges and the other the elimination of the hinge rotations (and therefore their couples) altogether by means of a new design principle. Some analysis of the latter scheme is provided including a prescription for choosing suitable design parameters.