WordPress query_var by domain

I'd like to add a query variable to all queries coming from a certain domain.

For example, mydomain.com and proxydomain.com both show the same WordPress site, but for users visiting via proxydomain.com, I'd like to be able to handle their queries differently.

Additionally, I'd like to apply some different CSS styles for visitors coming through proxydomain.com.

I was thinking I could check for the query_var and apply classes based on the presence of that variable.