How to draw spherical "caps" onto arbitrary axis using SphericalPlot3D?

I am trying to make a diagram for a project where there intersection of three shapes--two concentric spheres and one cone--is shaded or highlighted or in some way marked out. I currently have the following

But the mesh caps should be rotated pi/2 so that they are centered on the x-axis and not the z-axis. The code I am using to make these caps is

cap1 = SphericalPlot3D[6, {\[Theta], Pi/4, 0}, {\[Phi], 0,  2 Pi},
Boxed -> False, PlotPoints -> 50, ImageSize -> Large,
Boxed -> False, AxesOrigin -> {0, 0, 0},
PlotStyle -> {Directive[Gray, Opacity[0.2]],
Directive[Gray, Opacity[0.1]]},
PlotRange -> {{-12, 12}, {-12, 12}, {-12, 12}}]


And ideally I would like to connect them with a strip that lies on the surface of the intersecting cone, but I haven't figured out how to do this yet.

And suggestions would help. I have already had a go with ContourPlot3D and RevolutionPlot3D to no avail.

Thanks.

SphericalPlot3D[] is not spherically symmetric. The axis $$\theta=0$$ is special. But I guess you could rotate your cap:

Show[
Graphics3D[{Opacity[0.4], Sphere[{0, 0, 0}, 6]}],
MapAt[Rotate[#, {{0, 0, 1}, {1, 0, 0}}] &, cap1, 1]
]


• We can use the implicit form of the cone region x >= Tan[π/4] Sqrt[y^2 + z^2]]
cap1 = SphericalPlot3D[6, {θ, 0, Pi}, {ϕ, 0, 2 Pi},
Boxed -> False, PlotPoints -> 80, MaxRecursion -> 4,
ImageSize -> Large, Boxed -> False, AxesOrigin -> {0, 0, 0},
PlotStyle -> {Directive[Gray, Opacity[0.2]],
Directive[Gray, Opacity[0.1]]},
PlotRange -> {{-12, 12}, {-12, 12}, {-12, 12}},
RegionFunction ->
Function[{x, y, z, θ, φ},
x >= Tan[π/4] Sqrt[y^2 + z^2] ||
x <= -Tan[π/4] Sqrt[y^2 + z^2] ||
z >= Tan[π/4] Sqrt[x^2 + y^2]]]


• Or use ClipPlanes
cap1 = SphericalPlot3D[6, {θ, 0, Pi}, {ϕ, 0, 2 Pi},
Boxed -> False, PlotPoints -> 50, ImageSize -> Large,
Boxed -> False, AxesOrigin -> {0, 0, 0},
PlotStyle -> {Directive[Gray, Opacity[0.2]],
Directive[Gray, Opacity[0.1]]},
PlotRange -> {{-12, 12}, {-12, 12}, {-12, 12}}];
Graphics3D[cap1[[1]], ClipPlanes -> {{1, 0, 0, -3 Sqrt[2]}}]