Is there any difference between 相濡以沫 and 以沫相濡?
Also, is this kind of Chengyu (the first and last two characters are flipped) common? 


No, it's not common. 

相濡以沫 is the most people would use. 以沫相濡 is not that common, though we can understand it. 

We don't usually flip the character order in idioms.