The ductility and strength of spider draglines means that they outperform the best synthetic fibres, but surprisingly little is known about the torsional properties of this remarkable filament. Unlike a mountain climber swinging from a rope, a spider suspended from its silk thread hardly ever twists. Here we show that a spider dragline has a torsional shape 'memory' in that it can reversibly and totally recover its initial form without any external stimulus; its observed relaxation dynamics indicate that these biological molecules have successively different torsional constants.