Laminates made of composite anisotropic layers have a large varieties of different possible elastic responses. Still remaining in the classical field of linear elasticity, it is possible to obtain, with laminates, elastic behaviors that are very strange and completely unusual not only for classical materials, like metallic alloys, but also for the same laminates, when one bounds to consider their usual applications and stacking sequences. We show in this paper some strange combinations of plies giving rise to laminates possessing unusual elastic responses. The aim of the paper is threefold; first of all, to show that, contrary to what commonly thought, linear elasticity can still surprise, especially when anisotropy and heterogeneity enter the game. Second, that designers of composite structures can perhaps find a wealth of unexpected solutions that can be very interesting in some cases: it is sufficient to abandon the very restrictive rules commonly used in laminates design. Third, that an appropriate mathematical method, in this case, the polar formalism of Verchery, can open the way to new results, otherwise almost impossible to be found. Copyright © 2012 John Wiley & Sons, Ltd.