We are interested in the high-order approximation of anisotropic advection-diffusion problems on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to enjoy a discrete entropy structure, ensuring that the discrete long-time behaviour of solutions mimics the PDE one. The nonlinear scheme is designed so as to enforce the positivity of discrete solutions. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.