  It is known that to every 1|1 dimensional supercurve X there is associated a
dual supercurve \hat{X}, and a superdiagonal \Delta in their product. We
establish that the categories of D-modules on X, \hat{X}, and \Delta are
equivalent. This follows from a more general result about D-modules and purely
odd submersions. The equivalences preserve tensor products, and take vector
bundles to vector bundles. Line bundles with connection are studied, and
examples are given where X is a superelliptic curve.
