You can turn a keg upside down if you swap the tubes. Also, as long as the dip tube is completely covered in beer, and the gas in tube is clear of the beer, you should be able to serve.
There may be issues with clarity, since the keg normally does a good job of allowing sediment to settle below the dip tube, but that can be remedied as well by filtering, or jumping a clear beer into a new keg.
So, yes, if you are careful and know what you're doing you can nearly lay a keg on it's side and serve. I'm guessing you have a cooler that is sized so that you can lay kegs on their sides and stack them or something?