MOST materials shrink laterally and become less dense when stretched. Materials that both expand laterally (that is, have negative Poisson's ratio) and densify when stretched are of interest both from the fundamental and the practical points of view1-5. A few monocrystalline phases with negative Poisson's ratio are known3,4, but these do not densify when stretched. Here we present molecular-mechanics calculations for some hypothetical phases of carbon which exhibit both kinds of behaviour. The properties derive from the presence of bonds that act as hinges in extended helical chains. Other unusual properties of these phases include negative thermal expansion, dopant-controlled porosity and low-temperature polymorphism. Such structures can be envisaged for polyacetylene, polydiacetylene, polyphenylene and (BN)x phases, as well as for variants of some known, structurally related inorganic phases.