  The s-wave nucleon-nucleon (NN) scattering matrix ($S$-matrix) exhibits UV/IR
symmetries which are hidden in the effective field theory (EFT) action and
scattering amplitudes, and which explain some generic features of the phase
shifts. These symmetries offer clarifying interpretations of existing pionless
EFT expansions, and suggest starting points for novel expansions. The
leading-order (LO) $S$-matrix obtained in the pionless EFT with scattering
lengths treated exactly is shown to have a UV/IR symmetry which leaves the sum
of s-wave phase shifts invariant. A new scheme, which treats effective range
corrections exactly, and which possesses a distinct UV/IR symmetry at LO, is
developed up to NLO (next-to-LO) and compared with data.
