I think "Habsburg" is a good enough descriptor. It encompasses the Catholic, northern-influenced culture (if not actually in the north like Trieste) as well as the baroque, counter Reformation-derived artistic styles.

They had a couple of notable 20th century leaders.

LOL good point but I don't think that's it, although maybe whatever reason why I like these places is part of the reason Nazis wanted them :)

I think Stalin ruled them all briefly too.

I have been to Vienna, and it’s an amazing city, despite some of its awful rulers.

see the movie The Sound of Music