I arrived at this question by way of investigating knots, particularly the idea of "turning them inside out", in a sense. I called these their "complements". I noticed that the Borromean Rings (technically a link, but knot for my purposes) and the prime knot of four crossings were both equivalent to themselves when they had been turned inside out. Soon after that, I realized that the Borromean Rings corresponded to a wheel graph: $W_3$, and that in general, a knot that corresponds to $W_n$ will be its own complement. Not long afterwards, I figured out that finding which knots are their own complements is the same thing as finding the multigraphs that are their own duals.
Is there some sort of characteristic of a multigraph that I can use to make it easier to determine if it is its own dual? Or perhaps there is a finite list of families (one of which would be $W_n$) of multigraphs that are their own duals? Other relevant information would also be appreciated.
My understanding is that self-dual graphs are completely classified. The 3-connected planar graphs have a unique planar embedding and a unique dual. Archdeacon and Richter constructed all self-dual 3-connected graphs:
D. Archdeacon, R. Richter, The construction and classification of self-dual polyhedra. Journal of Combinatorial Theory, Series B 54, 37-48 (1992). http://www.sciencedirect.com/science/article/pii/0095895692900656
For lower connectivity, you have to be a bit careful about what you mean by self-dual. There are three notions: map self-dual, graph self-dual, and matroid self-dual. This paper discusses all of these notions and gives constructions for all self-dual graphs of lower connectivity:
B. Servatius, H. Servatius, Self-dual graphs. Discrete Mathematics 149, 223-232 (1996). http://users.wpi.edu/~bservat/self3.pdf