  This paper describes a Dehn surgery approach to generating asymmetric
hyperbolic manifolds with two distinct lens space fillings. Such manifolds were
first identified
  in work of Dunfield-Hoffman-Licata as the result of a computer search of the
SnapPy census, but the current work establishes a topological framework for
constructing vastly many more such examples. We introduce the notion of a
``jointly primitive'' presentation of a knot and show that a refined version of
this condition ``longitudinally jointly primitive'' is equivalent to being
surgery dual to a $(1,2)$--knot in a lens space. This generalizes Berge's
equivalence between having a doubly primitive presentation and being surgery
dual to a $(1,1)$--knot in a lens space. Through surgery descriptions on a
seven-component link in $S^3$, we provide several explicit multi-parameter
infinite families of knots in lens spaces with longitudinal jointly primitive
presentations and observe among them all the examples previously seen in
Dunfield-Hoffman-Licata.
