"....one can have the origin at the centre of the Earth, and the Z-axis along the polar axis. A person standing at the North Pole is at position (0, 0, R), at the South Pole (0,0, -R) where R is the Earth's radius.
A coordinate transformation from (0,0,R) to (0,0, -R) is equivalent to an upside down reflection
If you are travelling in a plane at a constant velocity you are in an plane's inertial frame of reference not the Earth's frame of reference. Your position and velocity is relative to the plane's frame of reference.
If your upside down it means the plane has turned upside down. 
"..
You have defined a special set of conditions, namely a frame of reference with the origin 0,0,0 at the earth.
Consider the universe: there are a very large number (some would say infinite) locations for the origin of a frame of reference. Those locations within the earth, are by comparison an almost infinitesimally small fraction. Choosing the centre of the earth as the origin is a very special case. In the general case my statement is true: there is no up or down.
Steven[/quote]
[quote=NotPrinceHamlet;432570]What I mean by euclidean space is just flat 3d space, with all 3 axes orthogonal to each other and all uniformly scaled - then even if the ant doesn't realise that it is upside down at the south pole, relative to the north pole, it is.