As we reach the end of Moore s law, digital logic uses irreversible logic gates whose energy consumption has been scaled toward a lower limit. Reversible logic gates can provide a dramatic energy-efficient alternative, but rely on reversible dynamics. Here, we introduce a set of superconducting reversible gates that are powered alone by the inertia of the digital input states, contrasting existing adiabatic prototypes which are powered by an external adiabatic drive. The classic model of an inertia-powered reversible gate uses ballistic particles which scatter in two dimensions, where the digital state is represented by the particle path. Our ballistic gates use as the bit state the topological charge (polarity) of a fluxon moving along a long Josephson junction (LJJ) such that the particle path is confined to one dimension. The fundamental structures of our reversible fluxon logic (RFL) are one-bit gates which consist of two LJJs connected by a circuit interface that comprises three large-capacitor Josephson junctions (JJs). Numerical simulations show how a fluxon approaching the interface under its own inertia converts its energy to an oscillating evanescent field, from which in turn a new fluxon is generated in the other LJJ. We find that this resonant forward scattering of a fluxon across the interface requires large capacitances of the interface JJs because they enable a conversion between bound-evanescent and traveling fluxon states (without external power). Importantly, depending on the circuit parameters, the new fluxon may have either the original or the inverted polarity, and these two processes constitute the fundamental identity and NOT operations of the logic. Based on these one-bit RFL gates, we design and study a related two-bit RFL gate which shows that fluxons can exhibit conditional polarity change. Energy efficiency is accomplished because only a small fraction of the fluxon energy is transferred to modes other than the intended fluxon. Simulations show that over 97% of the total fluxon energy is preserved during gate operations, in contrast to irreversible gates where the entire bit energy is consumed in bit switching. To provide insight into these phenomena, we analyze the one-bit gate circuits with a collective-coordinate model which describes the field in each LJJ as a combination of fluxon and mirror antifluxon. This allows us to reduce the many junction circuit (the three interface JJs and the many JJs approximating the LJJs, solved numerically) to that of two coupled degrees of freedom that each represent a particle. The evolution of the reduced model agrees quantitatively with the full circuit simulations and validates the use of the mirror-fluxon ansatz. Parameter tolerances are calculated for the proposed circuits and indicate that RFL gates can be manufactured and tested.