  We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic
problems that is capable of handling general meshes. The method hinges on
discrete unknowns that are polynomials attached to the faces and cells of a
coarse mesh; those attached to the cells can be eliminated locally using static
condensation. The main building ingredient is a reconstruction operator, local
to each coarse cell, that maps onto a fine-scale space spanned by oscillatory
basis functions. The present HHO method generalizes the ideas of some existing
multiscale approaches, while providing the first complete analysis on general
meshes. It also improves on those methods, taking advantage of the flexibility
granted by the HHO framework. The method handles arbitrary orders of
approximation $k\geq 0$. For face unknowns that are polynomials of degree $k$,
we devise two versions of the method, depending on the polynomial degree
$(k-1)$ or $k$ of the cell unknowns. We prove, in the case of periodic
coefficients, an energy-error estimate of the form
$\left(\varepsilon^{\frac{1}{2}}+H^{k+1}+(\varepsilon/H)^{\frac{1}{2}}\right)$,
and we illustrate our theoretical findings on some test-cases.
