The symbolism of the habit is “don’t look at me, I am nothing, look towards God” Hence the mask and the ‘verticality’ of the habit.
And yes, it probably is where the KKK got their idea, but that habit has been common in Spain and Portugal for centuries prior.
There is a building here in Detroit that was built in 1925. It is build in a pre Art Deco\Indian style. One of the features is a floor mosaic of themes from the local Native Americans, central to that was their symbol for the 4 winds. We would call that symbol a swastika. But, back in 1925, it was a simply an interesting design to most Detroiters reminicent of the local Wyandot tribes.
It’s been covered up by a rug since WW-II.