Name for geometric objects based on sphere / ball

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I looked in Wikipedia, Wolfram etc. but could not find an answer. If you take a spherical cone and mirror it at the apex to get something like a double-cone plus caps, does this have a special name? The remainder of the ball is something like a hybrid of toroid and spherical wedge. Is there a name for such a 3D-object or its 2D-analog, a 2-sphere minus two equal sized caps on the poles?