We investigate order isomorphisms, which are not assumed to be linear, between complete order unit spaces. We show that two such spaces are order isomorphic if and only if they are linearly order isomorphic. We then introduce a condition which determines whether all order isomorphisms on a complete order unit space are automatically affine. This characterisation is in terms of the geometry of the state space. We consider how this condition applies to several examples, including the space of bounded self-adjoint operators on a Hilbert space. Our techniques also allow us to show that in a unital C*-algebra there is an order isomorphism between the space of self-adjoint elements and the cone of positive invertible elements if and only if the algebra is commutative.