Reminder that the Nazis never called their symbol a "Swastika" and that it has nothing to do with the Hindu or Buddhist symbol properly called a Swastika

It literally is a coincidence. The Nazis called their symbol a Hakenkreuz. Hakenkreuz means "hooked cross", and like the name implies, comes from the Christian cross. The first proto-Nazis to use this were the Order of the New Templars, an antisemitic fascist organization that contributed to the rise of the Nazi party. Before that, it was a generic variety of Christian cross that was used for church decoration.

Now, it is true that when Indo-European artifacts containing the shape were found, some of which may be related to the Indic Swastika, that some Nazis said the two were one and the same, but that's different from the usual narrative of "Hitler stole it from India"

The truth is, Hitler didn't steal it, because he didn't even invent it. Surprise surprise, Mein Kampf is full of lies!!! Why would you take Hitler's word at face value???

and the thing is, the "Crux Gammadion" (the name I'm using for the shape as a whole, as its a descriptor of the shape; a cross made of 4 gammas) is found in so many cultures throughout the world, but not because they share some secret unified origin. No it's just an easy to make shape. It's like the triangle, star, or square. It's a shape people independently come up with.

This DOES NOT MEAN you can go willy-nilly "reclaiming it" though. As a Taoist who has explored Buddhism and is still involved in East Asian religions, I do have some religious stuff that use the symbol, but I'm very cautious with my usage of such and try to be courteous. I do not think there is any reason for European Neopagans to "reclaim" the shape and any attempt to do so is very suspect. I really don't think anyone should be reclaiming it if they're European and/or Christian.

Also the idea that Buddhist Swastikas always face left is wrong. Both orientations appear in Buddhist contexts all the time

Now why is there so much unsourced pop-history claiming Nazis stole it from India?
One is that it makes a nice little racial-separatist story about appropriation and why cultures shouldn't intermingle. Another reason is to push the blame of Nazism away from extremist Christianity. It's easy to go "The Nazis were a bunch of weirdo Pagan Hindus" instead of "The Nazis were everyday Germans who were "good Christians". This is why we see so much stuff about Nazi pagans, even though that was literally just Himmler and even Hitler didn't really like it, and not enough about how the bulk of Nazis were Christians or Atheists, the two belief groups that make up the far right today.