  In the second, fourth and fifth authors' previous work, a duality on generic
real analytic cuspidal edges in the Euclidean 3-space $\boldsymbol R^3$
preserving their singular set images and first fundamental forms, was given.
Here, we call this an `isometric duality'. When the singular set image has no
symmetries and does not lie in a plane, the dual cuspidal edge is not congruent
to the original one. In this paper, we show that this duality extends to
generalized cuspidal edges in $\boldsymbol R^3$, including cuspidal cross caps,
and $5/2$-cuspidal edges. Moreover, we give several new geometric insights on
this duality.
