First of all, Jackson died sixteen years before the Civil War, so he was hardly a Confederate.
Secondly, he was a staunch unionist whose stance on South Carolina's attempt at nullification in 1832 paved the way for the later Unionists that neo-Confederates hate so much.
Thirdly, he was a radical and one of the founders of the Democrat party. America's true conservatives at the time were the Federalists and Whigs, not the Democrats.
This confusion is what happens when people start thinking that populism is "conservative."
The Whig Party took its name from the Whigs of England who opposed royal absolutism--because they were opposed to "King Andrew the First."
This is true. While Jackson was mostly supportive of much of the ideology that later came to define the Confederacy (agrarianism, financial and economic de-centralism), he would not abide secession. Jackson's position was the Constitutionally correct one: there is nothing in the US Constitution that guarantees a right of a state to secede (Article I Section 10 bars states from printing currency, negotiating treaties, declaring war, or imposing tariffs - which makes it pretty clear by implication that states aren't sovereign entities that can secede at will).
I'd also add that Lincoln was in the right to wage a war against secession to preserve the union. Where he erred both Constitutionally and politically was with the Emancipation Proclamation, which turned what was a war to preserve the union into a war over slavery. Many Americans who were willing to fight and die to preserve the union (rightly) resented having to do so to end slavery.
Thirdly, he was a radical and one of the founders of the Democrat party. America's true conservatives at the time were the Federalists and Whigs, not the Democrats.
It's meaningless to retroject the issues defining today's political parties, or even today's political spectrum, back 150 years. The issues that defined Democrats vs. Republicans in the 19th century have nothing to do with the issues defining those parties now. For instance, the defining issue of the Democrats since the mid 20th century has become an expanded welfare state. Neither party supported a welfare state in the 1800's because the notion wasn't on anybody's radar then. Where you stood on tariffs was probably the most important Democrat vs. Republican (or Federalist/Whig) issue of the time. While people still debate tariffs today, that's hardly the key issue that distinguishes Democrats and Republicans today. Jackson's "radicalism" consequently had nothing to do with the issues the define political radicalism today.