Forgot to mention they did have, nothing to do with F1 championship wise, an exhibition race for a couple of years which was meant to be F1 v Indy cars, at Monza (I think on the oval). Billed as Race of Two Worlds , aka Monzapolis, the Monza 500, it was not quite as planned with many F1 cars and drivers saying no thanks, particularly the first year but they tried it for two years, 1957 and '58. Second year was slightly better but not the show it was expected to be and fizzled out.
More details here:
https://en.wikipedia.org/wiki/Race_of_Two_Worlds