  We propose two Hybrid High-Order (HHO) methods for the incompressible
Navier-Stokes equations and investigate their robustness with respect to the
Reynolds number. While both methods rely on a HHO formulation of the viscous
term, the pressure-velocity coupling is fundamentally different, up to the
point that the two approaches can be considered antithetical. The first method
is kinetic energy preserving, meaning that the skew-symmetric discretization of
the convective term is guaranteed not to alter the kinetic energy balance. The
approximated velocity fields exactly satisfy the divergence free constraint and
continuity of the normal component of the velocity is weakly enforced on the
mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov
fluxes for pressure-velocity coupling: a Harten, Lax and van Leer (HLL)
approximated Riemann Solver designed for cell centered formulations is adapted
to hybrid face centered formulations. The resulting numerical scheme is robust
up to the inviscid limit, meaning that it can be applied for seeking
approximate solutions of the incompressible Euler equations. The schemes are
numerically validated performing steady and unsteady two dimensional test cases
and evaluating the convergence rates on h-refined mesh sequences. In addition
to standard benchmark flow problems, specifically conceived test cases are
conducted for studying the error behaviour when approaching the inviscid limit.
