We devise a multiscale Hybrid High-Order (HHO) method. The method hinges on discrete unknowns that are polynomials attached to the mesh faces and cells; those attached to the mesh cells can be eliminated locally using static condensation. The main building ingredient is a multiscale reconstruction operator that maps onto a fine-scale space spanned by oscillatory basis functions. The method handles arbitrary orders of approximation $k\geq 0$, and is applicable on general meshes. 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 $(\varepsilon^{\frac{1}{2}} + H^{k+1} + {(\varepsilon/H)}^{\frac{1}{2}})$. We illustrate the theoretical results on periodic and locally periodic test-cases.