The classical flexure problem of non-linear incompressible elasticity is revisited for elastic materials whose mechanical response is different in tension and compression---the so-called bimodular materials. The flexure problem is chosen to investigate this response since the two regions, one of tension and one of compression, can be identified easily using simple intuition. Two distinct problems are considered: the first is where the stress is assumed continuous across the boundary of the two regions, which assumption has a sound physical basis. The second problem considered is more speculative: it is where discontinuities of stress are allowed. It is shown that such discontinuities are necessarily small for many applications, but might nonetheless provide an explanation for the damage incurred by repeated flexure. Some experimental evidence of the possibility of bimodularity in elastomers is also presented.