In order to get a handle on this, one has to be able to put them self in the period on the timeline.
America was founded on slavery so racism was rampant. Fast forward to the Civil War. The north thought slavery was too much but the whole country was still racist by considering blacks inferior. Even Lincoln was a racist ...
"Our republican system was meant for a homogeneous people. As long as blacks continue to live with the whites they constitute a threat to the national life. Family life may also collapse and the increase of mixed breed bastards may some day challenge the supremacy of the white man." ... -Lincoln.
Thus, the political affiliation of the KKK charter members was of little significance as racism was pervasive countrywide. Racism didn't become a political party issue until enough outrage in the mid 1950s made civil rights a political priority.
Battle lines were drawn and came to a head in 1964. LBJ signed the civil rights bill while famously saying the dems have lost the south for the next generation. The reps nominated Goldwater who was against civil rights. MANY Americans swapped parties over this highly charged issue, Strom Thurmond is symbolic of the transition
"Thurmond switched parties because of his opposition to the 1964 Civil Rights Act, disaffection with the liberalism of the national party, and his support for the conservatism of the Republican presidential candidate Senator Barry Goldwater."
Martin Luther King was aware of the transitioning ethos. In his autobiography he stated the following about the republicans at their convention in 1964 ...
"The Republican Party geared its appeal and program to racism, reaction, and extremism. All people of goodwill viewed with alarm and concern the frenzied wedding at the Cow Palace of the KKK with the radical right."
This led to the southern strategy embraced by Nixon ...
"In American politics, southern strategy refers to methods the Republican Party used to gain political support in the South by appealing to the racism against African Americans harbored by many southern white voters."
I have way too much more evidence of right wing racism to itemize here but, you are correct, the righties have the lion's share of racists.
The right wing tends to be authoritarian. The more extreme one gets, the more superior they feel. Thus, they use their superiority complex to contrast from and oppress those they consider inferior. Notice how the righties have no problem throwing a group of people (Muslims for example) on the same bus of oppression dismissive of the innocent people they persecute.