1-Bridge Braids in Solid Tori (Berge-Gabai knots), dual knots, & knot exteriors Let $K$ be a knot in a solid torus. Combining results of Berge and Gabai, we know that if $K$ admits a solid torus surgery, then $K$ is a 1-bridge braid. Using Gabai's result, we can figure out what the surgery slope is. If a 1-bridge braid admits a non-trivial solid torus surgery, the surgery slope is (almost) always unique (there is a unique 1-bridge braid which admits 2 non-trivial solid torus surgeries; most refer to it as K(7,2,4) in (w,b,t) notation).
If $B = K(7,2,4)$, we know that $B$ is equivalent to its dual, $B'$.
What is known about the relation between a 1-bridge braid and its dual in general? Namely:
Question #1: Given a 1-bridge braid $K$ that admits a solid torus surgery, when is it equivalent to its dual?
Question #2: Is it known when, given 1-bridge braids $K_1$ and $K_2$ such that $K_1 \not \simeq K_2$, their exteriors (in the solid torus $\mathbb{D}^2 \times S^1$) are homeomorphic? 
I know Berge gives a necessary & sufficient condition for equivalence of knots, but I'm wondering if the answers to my above questions already exist in the literature in a more modern/non-G-pair-theoretic language. 
Thanks!
 A: For Question #1:
As you acknowledge, Berge already gives an answer to Question #1.  Using his notation, this is summarized in Table 2 of his paper "The knots in $D^2 \times S^1$ which have nontrivial surgeries that yield $D^2 \times S^1$".  Berge also already tells you how to translate from a G-pair to a standard braid form in his Lemma 2.2.  So really, you just need to do the legwork to get it into a more common presentations of the dual.
Alternatively, a non-G-pair exhibition of the pairs of dual Berge knots is given in the setting of their tangle quotients in Baker-Buck, "The classification of rational subtangle replacements between rational tangles".  Figures 3-6 show bandings (+isotopy) between tangles in plat form keeping track of the dual arc, starting from a tangle isotopic to the crossingless two cup tangle.
(In  [Baker-Buck], since they’re dual, I had combined types III and V into one family, though I really shouldn’t have.  With one exception, there’s no overlap of them (as noted in Table 2 of [Berge]).   Referring back to my "Surgery descriptions and volumes of Berge knots II" one finds that in Figure 5 of [Baker-Buck]:  Top row shows V to III,  Bottom row shows III to V.)
For Question #2:
If the exteriors of $K_1$ and $K_2$ are both homeomorphic to the manifold $X$, then either 


*

*(A) their meridians are on the same boundary component of $X$ in which case they are a pair of Berge-Gabai duals as in Question #1, or

*(B) their meridians are on different boundary components of $X$ in which case this has been partially addressed by Wu in "The classification of Dehn fillings on the outer torus of a 1-bridge braid exterior which produce solid tori".


In Theorem 3.6 of [Wu], Wu determines both the 1-bridge braids for which a filling of the outer torus boundary of the exterior produces a solid torus again and the slopes of these fillings.  What’s still missing is the determination of the dual knot the resulting solid torus: Is it necessarily a 1-bridge braid? If so, which one?
Note that if you drop the assumption of 1-bridge braid, situation (B) becomes much more varied.  You’re basically looking at two component links in lens spaces where each component is isotopic (individually) to a core of a Heegaard solid torus.  For example in $S^3$ any two component link of unknots has this property.
