Modifying the MapAt+GeometricTransformation
approach in this answer in the related Q/A linked by @Mr.Wizard:
ClearAll[filledLLP]
filledLLP[l_, o : OptionsPattern[{Graphics, ListLinePlot}]] :=
Graphics[MapAt[GeometricTransformation[#, ReflectionMatrix[{-1, 1}]] &,
ListLinePlot[Reverse[l, 3], FilterRules[{o}, Options[ListLinePlot]]], {1}][[1]],
FilterRules[{o}, Options[Graphics]],
AspectRatio -> 1/GoldenRatio, Axes -> True, PlotRange -> All]
Examples:
lists = {{{0, 0}, {1, 1}}, {{5, 0}, {6, 1}}};
llp0 = ListLinePlot[lists, ImageSize -> 400, BaseStyle -> Thick];
llpX = filledLLP[lists, Filling -> {1 -> {2}}, FillingStyle -> Yellow,
ImageSize -> 400, BaseStyle -> Thick];
Row[{llp0, llpX}, Spacer[10]]

filledLLP[{{{0, 0}, {1, 1}}, {{1, .2}, {4, .5}}, {{5, 0}, {6, 1}}},
Filling -> {2 -> {{3}, Opacity[1, Red]},
2 -> {{1}, Opacity[1, Yellow]}, 1 -> {{3}, Opacity[.25, Cyan]}},
ImageSize -> 400, BaseStyle -> Thick]

axisFlip
function for this as illustrated here: (18669) $\endgroup$