I'm not just talking about the obvious parallels; they're all over the place:
1) The Kyklopes, Polyphemos/The one-eyed Bible Salesman, Big Dan Teague
In the poem, Odysseus happens upon an isle that is wild, verdant, and lush with food. He ventures with some men to a cave. They wait for the cave's inhabitant to return, and prevail upon his hospitality. Instead he messily devours some of them. They remain for a day, and sharpen a log while waiting. When he returns, Odysseus gets him drunk and they put his eye out with the fire-hardened spike. The poem's actually fairly graphic at this point, though I don't know how realistically. I've never had the good fortune to watch as someone's eye gets stabbed out with a burning hot, pointy stick. Anyway, they escape, though they leave a few behind.
Big Dan instead gets them to serve him a free meal, and pay for the one he was eating when he decided to rob them (two meals, just as in the poem). Then he beats the crap out of them (saving Ulysses Everett McGill for last, just as Polyphemos promised Odysseus he'd be last) and steals their money and car.
2) The Sirenes
Well, this was obvious. They ended up on the rocks. In the movie at least. In the book, Odysseus has his men tie him to the mast and stop up their ears with wax so he can listen. Turns out their song was about... wait for it... the glories of the Trojan War. That's right; they were going to lure him to his death with a song about his own prowess. More on this later.
The malign goddess didn't really show up in the movie, but I think there was a bit of a reference. Kirke's thing (everyone has a "thing" they do) was to lure people in for a feast and turn them into pigs (à la Bavmorda, in [u]Willow[/u]). Those she really liked, she'd sleep with until she got bored with them, then turn them into something noble, like a lion or a wolf. Odysseus survives because Hêrmes gives him a potion.
Anyway, the movie's parallel is Delmar's (mistaken) belief that Pete's been turned into a toad.
Did you catch "Pappy" O'Daniel's first name? Menelaos, just like the red-haired king, husband to Helen, for whom Odysseus sailed off to war in the first place.
Actually, one of his epithets was "Menelaos Xanthos", which is why he's called the red-haired king. "Xanthos" means "blond", or "slave". So why do they call him red haired? Have you ever seen a blond Greek? Of course not, so he couldn't have been blond.
Um, have you ever seen a red headed Greek?
5) Man of Constant Sorrow
I didn't get this until I watched it tonight. Here's a sample of the lyrics.
I am a man of constant sorrow,
I've seen trouble all my days.
I bid farewell to old Kentucky,
the place where I was born and raised.
Your friends think maybe I'm a stranger,
my face you'll never see no more.
Ha! Get it? It's the whole theme of both poem and movie! He's a wanderer, having adventures. That's his idiom (his "thing"). Odysseus bid farewell to old Ithaka, where he was born and raised.
6) The Wife
Penélopê in the poem, Penny in the movie (ha!). He has to leave her behind, and she gets courted while he's away. To quote one of Everett's daughters, "He's a suitor!"
In both poem and movie, the wife welcomes this attention.
Penny's motives are clear. She's got seven daughters to look after, and Vernon T. Waldrop (the suitor) has good prospects.
Penélopê's motives are less clear. She takes joy at their presence, even though they're impoverishing her. This is clearly indicated by the dream she relates to Odysseus in his guise as a beggar (was it even a real dream, or was she testing him?). I think Homer was simply an astute observer of the human scene. Greek women were fairly circumscribed. Their fathers, husbands, and sons were technically in control of their lives (though this poem and others indicate that women held more power de facto than they were considered to de jure). Legally, a woman had no real control over her life. With the suitors around. Penélopê was more than just the wife waiting for Odysseus; she was the center of the plot. Without them, Odysseus could just come home and she'd be a minor point to wrap up at the end of the poem. Because she was sought after, she became important. She was a proving point to her husband and her son. She had to be fought for. She became known for her guile in holding the suitors off for years. More than just the waiting wife, she was special and had to be acknowledged.
7) The Blind Prophet
Odysseus has to travel to the underworld in order to speak to Tiresias, the blind prophet of Thebes. Ulysses happens upon a blind prophet after escaping from the penal farm. Both prophets give unwelcome news.
Tiresias tells Odysseus that he'll be away from home for years, and will have to travel MORE after getting home. The nameless prophet of the film says they won't get the treasure they seek, but they'll get another one, despite obstacles in their path.
8) Finally, the man himself
Odysseus was famed for his ability as a thinker, a tactician, a debater, rather than as a fighter (although, as a hero, he was necessarilly death incarnate when the poet got around to it). He could talk anyone's ear off, lie on the spot, come up with whatever story he needed to get someone to do what he wanted. That was his skill. He told a long (and subtly insulting) story to the king of the Phiaikians in order to get a ride home. When he gets home, he tells a lie to the first person he meets. Hell, he lies to EVERYONE he meets. Even after he's killed the suitors and slept with his wife (while dissembling to her about the time he spent with Kalypso, nudge nudge), he lies to his father. He's just a fast talking, likeable liar.
And so is Ulysses Everett McGill.
Neither had much control over the men nominally under their command. Both were vain.
All in all, the movie was incredibly well written. I also enjoyed the many Wizard of Oz references. The KKK dance was funny, too, especially the lyrics.
Yo hi yo hi yo hi yo
eenie meenie miny moe
Ah well. This has gone on long enough. I'll talk about the odd feminist/antifeminist tone of the Odyssey some other time.