Cellulose is the most familiar and most abundant strong biopolymer, but the reasons for its outstanding mechanical performance are not well understood. Each glucose unit in a cellulose chain is joined to the next by a covalent C-O-C linkage flanked by two hydrogen bonds. This geometry suggests some form of cooperativity between covalent and hydrogen bonding. Using infrared spectroscopy and X-ray diffraction, we show that mechanical tension straightens out the zigzag conformation of the cellulose chain, with each glucose unit pivoting around a fulcrum at either end. Straightening the chain leads to a small increase in its length and is resisted by one of the flanking hydrogen bonds. This constitutes a simple form of molecular leverage with the covalent structure providing the fulcrum and gives the hydrogen bond an unexpectedly amplified effect on the tensile stiffness of the chain. The principle of molecular leverage can be directly applied to certain other carbohydrate polymers, including the animal polysaccharide chitin. Related but more complex effects are possible in some proteins and nucleic acids. The stiffening of cellulose by this mechanism is, however, in complete contrast to the way in which hydrogen bonding provides toughness combined with extensibility in protein materials like spider silk.