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Reversible logic can provide lower switching energy costs relative to all irreversible logic, including those developed by industry in semiconductor circuits, however, more research is needed to understand what is possible. Superconducting logic, an exemplary platform for both irreversible and reversible logic, uses flux quanta to represent bits, and the reversible implementation may switch state with low energy dissipation relative to the energy of a flux quantum. Here we simulate reversible shift register gates that are ballistic: their operation is powered by the input bits alone. A storage loop is added relative to previous gates as a key innovation, which bestows an asynchronous property to the gate such that input bits can arrive at different times as long as their order is clearly preserved. The shift register represents bit states by flux polarity, both in the stored bit as well as the ballistic input and output bits. Its operation consists of the elastic swapping of flux between the stored and the moving bit. This is related to a famous irreversible shift register, developed prior to the advent of superconducting flux quanta logic (which used irreversible gates). In the base design of our ballistic shift register (BSR) there is one 1-input and 1-output port, but we find that we can make other asynchronous ballistic gates by extension. The gate constitutes the first asynchronous reversible 2-input gate. Finally, for a better insight into the dynamics, we introduce a collective coordinate model. We find that the gate can be described as motion in two coordinates subject to a potential determined by the input bit and initial stored flux quantum. Aside from the favorable asynchronous feature, the gate is considered practical in the context of energy efficiency, parameter margins, logical depth, and speed.