EDIT: A solution with Show and 3 Plots, but I would prefer a more elegant way to do this inside of one Plot command:
Manipulate[
Rotate[Show[
Plot[{a (\[Eta] - 1)^2}, {\[Eta], 0, 1},
PlotRange -> {{-1, 1}, {0, 1}},
PlotStyle -> {Thick, Dashed, ColorData[1, 2]}, AspectRatio -> 1.5],
Plot[{a (\[Eta] + 1)^2}, {\[Eta], -1, 0},
PlotRange -> {{-1, 1}, {0, 1}},
PlotStyle -> {Thick, Dashed, ColorData[1, 2]}, AspectRatio -> 1.5],
Plot[
{xL + 0.2*\[Theta][1][\[Eta], \[Xi], Nterms], xL}, {\[Eta], -1, 1},
PlotRange -> {0, 1}, Filling -> {1 -> xL},
PlotStyle -> {{Thick, ColorData[1, 1]}, {Thick, ColorData[1, 2]}},
AspectRatio -> 1.5]
], rotateAngle],
{\[Xi], 0, 0.1},
{xL, 0, 1},
{{a, 1}, 0, 1},
{{rotateAngle, -Pi/2}, {0, -Pi/2}}
]