  We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to
approximate elliptic interface problems. The curved interface can cut through
the mesh cells in a very general fashion. As in classical HHO methods, the
present unfitted method introduces cell and face unknowns in uncut cells, but
doubles the unknowns in the cut cells and on the cut faces. The main difference
with classical HHO methods is that a Nitsche-type formulation is used to devise
the local reconstruction operator. As in classical HHO methods, cell unknowns
can be eliminated locally leading to a global problem coupling only the face
unknowns by means of a compact stencil. We prove stability estimates and
optimal error estimates in the $H^1$-norm. Robustness with respect to cuts is
achieved by a local cell-agglomeration procedure taking full advantage of the
fact that HHO methods support polyhedral meshes. Robustness with respect to the
contrast in the material properties from both sides of the interface is
achieved by using material-dependent weights in Nitsche's formulation.
