I think you do not get reincarnation. I don't believe in it, but it is expressed different. Most important question that should be asked when discussing reincarnation is, what is being reincarnated?
When you talk of yourself being reincarnated, what is meant by self? Most westerners understand it as a "soul" being born again, and you being that soul. But that is not the idea in the slightest. Atman Is not you as a person, nor a personality. But rather this unmoved mover that keeps being reborn in every sentient being, the silent watcher.
If you bring the concept of Karma into this, it complicates things. The idea of Atman is older than the idea of Karma. Karma is a cheap moral one upmanship and cheap catharsis for people to deal with hardships of life. The actual Buddha and the Upanischad before him hardly ever talked about Karma and when they did, they didn't insert a metaphysical reality to it but rather a materialistic one.