  Recent literature posed the problem of characterising the graph degree
sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected)
realisation. This seems to be a difficult problem in full generality. In this
paper, we characterise the sequences with exactly one self-dual $3$-polytopal
realisation.
  An algorithm in the literature constructs a self-dual $3$-polytope for any
admissible degree sequence. To do so, it performs operations on the radial
graph, so that the corresponding $3$-polytope and its dual are modified in
exactly the same way. To settle our question and construct the relevant graphs,
we apply this algorithm, we introduce some modifications of it, and we also
devise new ones. The speed of these algorithms is linear in the graph order.
