We discuss a non-reversible, lifted Markov-chain Monte Carlo (MCMC) algorithm for particle systems in which the direction of proposed displacements is changed deterministically. This algorithm sweeps through directions analogously to the popular MCMC sweep methods for particle or spin indices. Direction-sweep MCMC can be applied to a wide range of reversible or non-reversible Markov chains, such as the Metropolis algorithm or the event-chain Monte Carlo algorithm. For a single two-dimensional tethered hard-disk dipole, we consider direction-sweep MCMC in the limit where restricted equilibrium is reached among the accessible configurations for a fixed direction before incrementing it. We show rigorously that direction-sweep MCMC leaves the stationary probability distribution unchanged and that it profoundly modifies the Markov-chain trajectory. Long excursions, with persistent rotation in one direction, alternate with long sequences of rapid zigzags resulting in persistent rotation in the opposite direction in the limit of small direction increments. The mapping to a Langevin equation then yields the exact scaling of excursions while the zigzags are described through a non-linear differential equation that is solved exactly. We show that the direction-sweep algorithm can have shorter mixing times than the algorithms with random updates of directions. We point out possible applications of direction-sweep MCMC in polymer physics and in molecular simulation.