What to call this kind of symmetry in a sphere?

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Geometrically, if the two hemispheres of a spherical distribution of some kind (let's say a spherical gas cloud) are similar such that the properties of the gas as seen by a person standing on a circle dividing the two hemispheres would see the same exact properties whether they looked up or down, does that kind of symmetry have a name? I would guess either equitorial symmetry or hemispheric symmetry but I don't know if those have different meanings than what I am trying to describe.