Uncensored2008
Libertarian Radical
You're confusing the Democratic Republicans (the one-time Jackson Republicans) as being Both Democrats and Republicans which are two very different ideologies. That party eventually became the Democratic Party while the National Republicans (the one-time Adams Republicans) became the Whig Party then finally the Republican Party.
Jackson never called his party Republicans.
The Democratic Republicans became the National Republicans when John Quincy Adams became President. Andrew Jackson and Aaron Burr then created the Democratic Party. While there was a common root, the split was severe and irreconcilable.
Jackson is, as you said, the founder of the democratic party. The "Trail of Tears" was the first great achievement of the democrats. They haven't changed much since.