Nope, not all had a rear swaybar. Actually, it is the rarity to have a rear bar. Only time you got a rear bar is if you ordered the F41 packages, what a Salon, Calais, 442 or HO had, or if you had a supreme/brougham and added the F41 as an option. If it has a 4 bolt quick ratio box, it may also have a rear bar. Dont know what that option code was but ive found a few like that. My 84 supreme parts car had a factory quick ratio box and rear swaybar.
My 81 supreme didnt have a rear bar when I got it, and neither did my 85 Salon.
If every cutlass you see has a rear bar, id be super suprised.
And that car reminds me of a local one, one of those "coach builder" cars. No rear windows, wierd options, ect. The trim could have been painted from the factory that color judging by the sticker on the back. It could be a poorly painted home custom though.