Except this is patently silly.
Firstly, there is no such thing as left/right/aka/etc.. It just means that one has not thought it through, though.
The across-space is that perpendicular to up/down and front/back [ie space divided by gravity and time]. In three dimensions, this is linear, giving rise to left and right. In four dimensions, this is a hedrid or 2d space. This means, that a single appendage, like an arm or a tail, can freely point to any direction of this hedrid, while still being perpendicular to both up/down and front/back.
This space has no grain. It is possible for several individuals to lie at any angle of this space.
Further, there is no particular reason for four directions here: six might as equally do.
What /does/ exist in four dimensions, is that there is a -direction-of-rotation- that is universal to all members. That is, even though the specific angles might not be namable, the notion of left-turn and right-turn still do.
The dream you dream alone is only a dream
the dream we dream together is reality.