If you think about it Boris there are two different situations.
1. Up and Down are properties of a gravity well so on earth there is an up and down. Its direction is independent of the orientation of the observer. There is also a different up and down on the moon but there is no up and down where there is no gravity.
2. Right, left, forwards and backwards are different. They are properties of the individual. I have a right, left forwards and backwards which I take with me wherever and in whatever orientation I am. It is different to your left, right, forward and backward and everyone else's however.