I've never really had any cause to think about this before, but both past tense forms seem natural to me, and I think I use them both. I'm from the UK, if that helps.
I was trying to think of examples when I'd use each one, and it felt to me that I'd use spun if someone spun another object, and span if it was themselves that had been doing the spinning, so e.g. "I span around" (I turned around quickly) vs "I spun the crank" (I caused the crank to spin).
That said, I can also provide counter-examples: "I span the gyroscope" and "I spun around" also both sound completely natural to me, just that I think that I personally would use the other word in both cases.
I was trying to think of examples when I'd use each one, and it felt to me that I'd use spun if someone spun another object, and span if it was themselves that had been doing the spinning, so e.g. "I span around" (I turned around quickly) vs "I spun the crank" (I caused the crank to spin).
That said, I can also provide counter-examples: "I span the gyroscope" and "I spun around" also both sound completely natural to me, just that I think that I personally would use the other word in both cases.