Two domains, one name?

So, here’s the thing.
I’m hosting a site on my server, let’s call it mydomain.com. In fact, let’s say the site’s name is thesite.net, so I’m actually hosting it on mydomain.com/thesite.

I’ve also bought the domain name (without serverspace) thesite.net which I redirect to mydomain.com/thesite

What I’d like to do, is to be able to change all permalinks from mydomain.com/thesite/thepermalink to thesite.net/thepermalink. Am I making any sense?

To simplify, I’d like the permalinks to point to another url completely. Is this at all possible?