They did that to me at least once. I found out when the customer asked me for some certification. It was infuriating, because I was the only one not lying: the customer wanted to present me to their clients as their employee, when I was actually a subcontractor, so they didn't even complain to my bosses.

I don't know what I would do in your position. I have almost 25 years of experience and won't lie ever. If I can't get some job, I'd look for another one worse paid.