Of course in reality none of that's entirely accurate, any more than it would be accurate to imply that a given political party today represents the same thing it did 50, 100, or 150 years ago --- which I know Dave would never do...
The KKK (the first one) was started by CSA Army veterans, not politicians, as were several other vigilante groups at the time that wanted to ignore surrender and continue the Civil War. Meanwhile the Democratic Party, being the largest and oldest and in the South (where a president from a then-upstart new party had just defeated them) in effect the
only party, represented very different factions in different regions. That was a time when a Liberal or Conservative might belong to either party, and in the South those conservatives were all Democrats. So "terrorist arm of the Democratic Party" is quite a stretch, implying some kind of natal relationship. For being a Democrat in Mississippi was a whole different ball game from being a Democrat in Massachusetts.
The Klan was gone and driven out by the end of the 1870s. The new Klan revived in 1915 by an ex-minister doctor named Simmons, did get involved in politics and actually elected a few governors and a Senator, who were all Republicans. This second Klan also got a Democratic governor (Walton) impeached when he tried to drive them out of Oklahoma.
So the KKK has always involved conservatives but political party has been whatever was convenient at the time. If there was any political party at all.