  Computational efficiency demands discretised, hierarchically organised, and
individually adaptive time-step sizes (known as the block-step scheme) for the
time integration of N-body models. However, most existing N-body codes adapt
individual step sizes in a way that violates time symmetry (and symplecticity),
resulting in artificial secular dissipation (and often secular growth of energy
errors). Using single-orbit integrations, I investigate various possibilities
to reduce or eliminate irreversibility from the time stepping scheme.
Significant improvements over the standard approach are possible at little
extra effort. However, in order to reduce irreversible step-size changes to
negligible amounts, such as suitable for long-term integrations of planetary
systems, more computational effort is needed, while exact time reversibility
appears elusive for discretised individual step sizes.
