  In this paper, we explore the relationship between integrability and the
discrete holomorphicity of a class of complex lattice observables in the
context of the Potts dense loop model and the O(n) dilute loop model. It is
shown that the conditions for integrability, namely, the inversion and
Yang-Baxter relations, can be derived from the condition of holomorphicity of
the observables. Furthermore, the Z-invariance of the models is shown to result
in the invariance of the observables on the boundary of a sublattice under
reshuffling of the rhombuses of its planar rhombic embedding.
