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Stress-strain constitutive relations in solids with an internal angular degree of freedom can be modelled using Cosserat (also called micropolar) elasticity. In this paper, we explore the phenomenology for a natural extension of Cosserat materials that includes chiral active components and odd elasticity. We calculate static elastic properties of such a solid, where we show that static response to rotational stresses leads to strains that depend on both Cosserat and odd elasticity. We then compute the dispersion of linear solutions in these odd Cosserat materials in the overdamped regime and find the presence of \emph{exceptional points} in the dispersion relation. We discover that these exceptional points create a sharp boundary between a Cosserat-dominated regime of complete wave attenuation and an odd-elasticity-dominated regime of propagating waves. We conclude by showing the effect of Cosserat and odd elastic terms on the polarization of Rayleigh surface waves.