No, actually (I was confused myself on the subject.)
Previously, all crossers in either direction received a new name...e.g. one that crossed from the Atlantic to Eastern Pacific was renamed with the next Eastern Pacific name, and vice versa.
However, that was changed in 2001, and Adrian will remain Adrian IF IT REMAINS a system while crossing Central America into the Atlantic. If it disappates and regenerates, it will be renamed Arlene.
We'll see. I just read in my local newspaper (Mobile Register) today that it will be re-named Arlene.