The Hamiltonian Monte Carlo method is well-known for its ability to generate distant proposals and avoid random-walk behaviour. Its sampling efficiency however is highly sensitive to the choice of the number of leapfrog integration steps. Although the No-U-Turn Sampler automates the tuning of this parameter, it is computationally expensive and practically challenging to implement, especially on parallel architectures. In this work, we introduce the Reflected Hamiltonian Monte Carlo sampler, an HMC methodology that builds upon a reflection mechanism also used in the Bouncy Particle Sampler. The algorithm has an update rate parameter that plays an analogous role to that of the number of leapfrog integration steps in Hamiltonian Monte Carlo. With a focus on high-dimensional classification tasks, we demonstrate the competitive performance of the proposed algorithm against well-tuned Hamiltonian-based Markov Chain Monte Carlo methods.