Janie,
Yes, the Singer 66-16 and 18 and later models has a reverse of sorts. They call it a back tacker.
The later 15s have a reverse.
The HA-1 15 clones have reverse
Many others as well.
With the exception of the 9W-7 all Singers rotate counterclockwise, towards you as you sew.
The 9W-7 rotates clockwise away from you as you sew.
The machine is not sewing in reverse and will jam up or create birds nests under the needle plate when you turn it the wrong way. So you need to use them correctly.
My 66s will break the thread as well when turned wrong. All this just makes it more important to know your machine and make sure you get it going the correct direction.
Joe