I think if you sketched up a concept and figured out how much material went into it the machine would be much heavier than a conventional machine. The basic frame structure would have to be L shaped. With a small machine such as you are considering this is probably no problem but with large industrial machines it could be a factor.
You are quite correct about the linear ways not caring about which way is 'up'. If you had a big reinforced concrete wall handy you could bolt a conventional machine to it and now you have it . Although it is likely that the weight of the machine itself would cause distortion leading to inaccurate operation. The bed of a lathe is machined sitting level, the casting has to sag a little between the supports but when it is installed and levelled for use the sag will be the same as when it was made. If you rotated it 90 degrees now the sag due to its weight will be perpendicular to the direction it was machined; the machine would be warped downward which is sideways from the conventional view. The original sag would turn into an upward bow which would be outward when looking at it hanging on the wall.