Tarkovski's movies have what you could call a Russian sentiment. Hard to put in words what exactly that means but perhaps some fatalism and melancholy, how things are what they are and do not easily change except in our dreams. Looking back at Soviet Union and now Putin's Russia, and even the autocratic Czarist Russia before revolution, it is easy to understand where that sentiment comes from. But it makes for great, thoughtful art.
Another great Russian movie from a contemporary director of Tarkovski is "A Few Days from the Life of I.I. Oblomov":
Well put. Fatalism and melancholy are great descriptions of it.
One such interesting culture clash happened when the Soviets went to Cuba to shoot https://en.wikipedia.org/wiki/I_Am_Cuba. A beautiful movie, fantastic cinematography by Kalatozov, but the Cubans just never liked it. The portrayal of Cubans is very much what Russians thought Cubans might be like, as opposed to how Cubans see themselves.
And that sentiment is replicated by the director Andrey Zvyagintsev, in e.g. The Return (2003) or Leviathan. Both great movies with beautiful cinematography and very Russian feeling.
Another great Russian movie from a contemporary director of Tarkovski is "A Few Days from the Life of I.I. Oblomov":
https://www.imdb.com/title/tt0079619/