We present both hp-a priori and hp-a posteriori error analysis of a mixed-order hybrid high-order (HHO) method to approximate second-order elliptic problems on simplicial meshes. Our main result on the hp-a priori error analysis is a 12-order p-suboptimal error estimate. This result is, to our knowledge, the first of this kind for hybrid nonconforming methods and matches the state-of-the-art for other nonconforming methods as discontinuous Galerkin methods. Our second main result is a residual-based hp-a posteriori upper error bound, comprising residual, normal flux jump, tangential jump, and stabilization estimators (plus data oscillation terms). The first three terms are p-optimal and only the latter is 12-order p-suboptimal. This result is, to our knowledge, the first hp-a posteriori error estimate for HHO methods. A novel approach based on the partition-of-unity provided by hat basis functions and on local Helmholtz decompositions on vertex stars is devised to estimate the nonconformity error. Finally, we establish local lower error bounds. Remarkably, the normal flux jump estimator is only 12-order p-suboptimal, as it can be bounded by the stabilization owing to the local conservation property of HHO methods. Numerical examples illustrate the theory.