The first time I read it was a good while ago, probably 2003, give or take a year. Maybe it was denorae who recommended it to me? It was my first exposure to Calvino. Since that time I've read Baron in the Trees (loved it) Cosmicomics (couldn't finish it, didn't really grab me) and, of course, If on a Winter's Night a Traveler (just a little too twee and meta, and definitely makes an uncomfortable lot of assumptions about the reader, but still kind of cute in its way)
But of course I find it easy to project my own personal history of interrogating a writer's works on the works themselves, and in that way Invisible Cities is clearly the Calvino ur-book, from which all the rest are derivations. But being as it is short (so short! typically one or two pages) stories, prose poems, really, his style does feel very concentrated, cooked down to the essentials. It's a love letter to cities, the way cities are composed of people, and the way that cities are individuals, have personalities and repressed emotions --- and so that resonates a bit with me, but the clever little thematic rhythm that the wikipedia page spells out does nothing for me. It doesn't have any feeling of narrative, no conflict that's resolved or deep question that's answered. Maybe I shouldn't be looking for that in a Wunderkammer of weird little urban magical realism blurbs, but somehow I still want it and don't get it.