Owing to the complexity of geological features, the simulation of elasto-acoustic wave propagation in the Earth, is challenging. To handle efficiently this difficulty, we consider hybrid high-order (HHO) methods, which offer several attractive assets such as local conservativity, geometric flexibility through the support of polyhedral grids, and high-order precision [Di Pietro, Ern, 2014, Comput. Meth. Appl. Mech. Engrg.]. HHO methods rely on a pair of unknowns, combining polynomials attached to the mesh faces and the mesh cells. The cell unknowns can be eliminated locally using a static condensation procedure, whence an increased computational efficiency with respect to classical DG methods. HHO methods are closely related to hybridizable DG (HDG) methods [Cockburn, Gopalakrishnan, Lazarov, 2009, SINUM]. Here, we build upon previous work on HHO methods for either elastic or acoustic wave propagation in the time domain [Burman, Duran, Ern, 2022, Comm App Math Comp Sci], and devise the coupling between both HHO discretizations. A first-order formulation in time is considered, and implicit and explicit Runge--Kutta schemes are used for the time discretization. An energy balance confirming the stability of the schemes is derived. Numerical results on test cases with analytical and semi-analytical solutions show that the methods deliver optimal convergence rates. A more realistic case on the propagation of an elastic pulse in a sedimentary basin coupled to the atmosphere is presented. In accordance with what is observed in practice, we show that in such a case a significant part of the energy is captured by the sedimentary basin and then transmitted to the atmosphere.