Surface-knot is an embedded surface in $\Bbb{R}^4$. Project the surface in $\Bbb{R}^3$ gives the surface diagram with set of singularity points consists of double points, triple points and branch points. The closure of the preimage of double points is called double decker set. It is divided into two parts: upper decker set and lower decker set.
My question is: what is the relationship between upper decker set and lower decker set in the surface knot? Can we define an involution sending upper decker set to lower decker set. In other word, can we get the lower decker set from the upper decker set and conversely, if so how?
The simplest example when the double decker set consists of immersed circles in the surface (ribbon knots). Let the lower decker set Sb be immersed circles on a surface $F$. How about the upper decker set $S_a$? Is there any relationship between $S_a$ and $S_b$ in $F$?
Thank you in advance.