The asymmetry exists because your starting points are different by a factor of 1,000, so percentage changes are 1,000 times different in absolute terms.
A rise from $1 to $1000 is a 99,900% rise, but a fall from $1000 to $1 is only a 99.9% fall. So in percentage terms, the profit is symmetrical. A long position where the stock rises 10% will have a 10% gain. A short position where the stock falls 10% will have a 10% gain.
It has nothing to do with long and short, just an asymmetry between percentage change and absolute change. If you were long $1,000 on two stocks, one went from $1 to $1,000, and the other went from $1,000 to $1, you'd have a 99,900% gain on the former and a 99.9% loss on the latter.