Many continuous reaction-diffusion models on $\mathbb{Z}$ (annihilating or coalescing random walks, exclusion processes, voter models) admit a rich set of Markov duality functions which determine the single time distribution. A common feature of these models is that their generators are given by sums of two-site idempotent operators. In this paper, we classify all continuous time Markov processes on $\{0,1\}^{\mathbb{Z}}$ whose generators have this property, although to simplify the calculations we only consider models with equal left and right jumping rates. The classification leads to six familiar models and three exceptional models. The generators of all but the exceptional models turn out to belong to an infinite dimensional Hecke algebra, and the duality functions appear as spanning vectors for small-dimensional irreducible representations of this Hecke algebra. A second classification explores generators built from two site operators satisfying the Hecke algebra relations. The duality functions are intertwiners between configuration and co-ordinate representations of Hecke algebras, which results in a novel co-ordinate representations of the Hecke algebra. The standard Baxterisation procedure leads to new solutions of the Young-Baxter equation corresponding to particle systems which do not preserve the number of particles.