At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?

What is the strongest, most impressive statement one can make here? The Borsuk-Ulam Theorem applies, but I am uncertain of its full implications. Could one say that the two points are (1) separated by a specific geodesic distance, (2) have the same temperature, and (3) have the same barometric pressure? For example...? I pose this question for its pedagocial import, but it clearly follows from known theorems.

To what extent do these results extend to $\mathbb{R}^d$ for $d>3$?
Thank you for your help!

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