3D graphics showing a double symmetric hemispherical sector diagram

Question

I never used Mathematica to plot something that can be considered non-trivial. I want to create a plot similar to the one used here
on the left, but with an equal (symmetric) cone on the bottom hemisphere, with a plane cutting the two hemispheres creating the two equal opposite spherical caps, and perhaps using some nice translucent effects. I am a bit lost. Could you please suggest any reference/link that I can study to achieve my goal?

Daniel Huber · Accepted Answer

Something like;
theta = Pi/4;
Graphics3D[{Opacity[0.2],
  Sphere[],
  Cone[{{0, 0, Cos[theta]}, {0, 0, 0}}, Sin[theta]],
  Cone[{{0, 0, -Cos[theta]}, {0, 0, 0}}, Sin[theta]],
  Cylinder[{{0, 0, -10^-6}, {0, 0, 10^-6}}, 1]
  }]

3D graphics showing a double symmetric hemispherical sector diagram

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