Grade three perfect ideals and length four self-dual resolutions

Starting with a grade three perfect ideal $I\subset R$, we demonstrate how to produce the a self-dual resolution of length four using the resolution of the original ideal. This process is also reversible. The main case of interest is when the grade three perfect ideal has type two, so the output complex resolves $R\mathrm{/}J$ for a grade four Gorenstein ideal $J$. This suggests that the structure theory of these two families of ideals should be closely related.