  In this work, we develop a fully implicit Hybrid High-Order algorithm for the
Cahn-Hilliard problem in mixed form. The space discretization hinges on local
reconstruction operators from hybrid polynomial unknowns at elements and faces.
The proposed method has several assets: (i) It supports fairly general meshes
possibly containing polygonal elements and nonmatching interfaces, (ii) it
allows arbitrary approximation orders, (iii) it has a moderate computational
cost thanks to the possibility of locally eliminating element-based unknowns by
static condensation. We perform a detailed stability and convergence study,
proving optimal convergence rates in energy-like norms. Numerical validation is
also provided using some of the most common tests in the literature.
