Mugwump

I haven't heard this phrase in awhile and was
curious of it.

"...mug on one side whump on the other."

Today I saw the words spoiled mugwhumps in
Charlie Daniels'
- Open Letter To The Hollywood Bunch,
using the word "mugwhumps."

What exactly is a mugwhump?

That must be a misspelling of "mugwump." American Heritage Dictionary:
1. (Often capital M.) A Republican who bolted his party in 1884, refusing to support James G. Blaine as candidate for the U.S. presidency. Any person who acts independently, especially in politics. [Natick "mugquomp," "mugwomp," "captain."]