You know, for "not being a proper member of the pantheon," and fro being "primarily the antagonist," Loki took a lot of crap 'for the team' with out anything in the way of reward, and frequently with ridicule. Ya might want to take a good, hard look at the histories. Yeah, he was a trickster, and yeah he got bored a lot, but there were a number of times he saved the day and either got reamed, raped or beaten for his trouble with out the smallest thanks. Yeah, sounds like a real antagonist to me... Sounds more like the party clown to me... Yeah, your stuff ain't safe around him, but he's the ONLY one that gets to mess with you.
To answer the question, no, Loki the Red has no official symbol, icon or logo.