Strangely enough, it was a popular thought in the 19th century. The thinking was that the 'warrior' Aryans were a migratory people who established themselves as ruling castes over 'inferior' groups, but 'degenerated' by intermarriage with the locals in most places.

They thought that there was a shared ancestry. The generally thinking was that the genes did not get as "diluted" moving westward to Europe but did get mixed with indigenous peoples in India accounting for those differences.

Theoretically, those shared ancestors saw the swastika as an auspicious symbol but it was those that migrated to India that wrote about it first. Which is why its commonly referred to by its Sanskrit name.