It’s almost too obvious to say that the Homeric epics are misogynist. What strikes me is how systematically misogynist they are, how they have to keep repressing a feminine element that always threatens to resurface. Sometimes it does, most dramatically when Hecuba bares her breast in an attempt to dissuade Hector from rejoining the battle, more mundanely when the soldiers chide each other for being womanly.
Every story that is, on the face of it, about the relationship between a man and a woman is displaced into a story about a rivalry between two men or about the loyalty between two men. The presenting issue in the Iliad combines both moves: Helen is induced to be unfaithful to her rather unimpressive husband, Menelaus, but this marital tension is displaced into a rivalry between Menelaus and Paris (abortively staged in Book 3) and then of course balloons into an international conflict lasting a decade. Achilles’ apparently sincere love for his war captive, Briseis, quickly becomes fodder for rivalry with Agamemnon. This does not explode into violence due to party loyalty, but Achilles can only join back into the captive after the death of his beloved comrade Patroclus (male loyalty) opens up a more serious male rivalry with Hector. That then sets up the uneasy truce between Achilles and Priam — where two men, united in their grief over men, call a temporary halt to the war started over a woman.
This aggressively homosocial gender politics may work in war, but it starts to fall apart when we turn back toward home in The Odyssey. And that incoherence is brought to a head in the figure of Penelope. On the one hand, Homer seems to be trying to set up some kind of tension through the constant reminders of Agamemnon’s fate — will Penelope really be faithful, or will Odysseus be betrayed? On the other hand, there are only two ways for a human woman to be: either utterly devoted and submissive or maliciously traitorous. (Helen shuttles back and forth, but at any given moment she is either one or the other.) At times this strains credulity, as when we learn how happy Briseis was at the prospect of becoming the lawfully wedded wife of her husband’s murderer, or what Penelope exults after Telemachus basically tells her to shut up because he’s the man.
There is no room for a woman who is seriously torn, though Penelope sometimes come close. Unable to give her a complex internal life, Homer instead puts her into a complex situation where she can’t be sure whether her husband is dead or alive. If we ask why she doesn’t simply tell the Suitors to leave, the simplest answer is probably that if she did that, she wouldn’t be a proper submissive woman — open wilfulness, even in the service of faithfulness to her husband, is breaking the rules. Her only weapon is passive-aggression, exemplified by the burial shroud trick.
In the later epic tradition, it is the women who start to get what we moderns would identify as a complex interior life — above all the impressive figures of Medea and Myrrha, who are faced with a genuine internal conflict (whether to betray her father for Jason and whether to seduce her own father, respectively). As for the men, we see profound depths of emotion — to a point that is almost comical from a modern perspective at times, or at least from the perspective of impatient student readers who are tired of all the crying — but never real depth of character. The price they pay for their relentless repression of the feminine is being stuck at the surface of things.
An und für sich 
I wonder if this opposition (complex interior life for female characters/surface emotion for male ones) is related to the fact that (upper-class) women were confined to the home in Athens? The female essentially is the interior whereas the male is essentially public.
I think that’s right. One could even hypothesize that this was the deep connection that drew Freud so inexorably toward Greek illustrations in developing his theory about upper-class women confined to the home.
I don’t know if it is of interest: what about the goddesses (Athena or Aphrodite, I think) who take part in battles in the Iliad? Also, Medea was a sorceress, Calypso and Circe had quasi-supernatural powers, Cassandra was prophetess. What about these female figures that stand at the limit?
That is a really interesting point. It’s as though there’s something divine in transcending conventional femininity (or its “evil” obverse) — though whether we take it as “she’s a goddess, so it’s okay” or “no mortal woman could be strong enough to transcend that limit” is an open question. One could even say there’s something a little divine about Helen, allowing her to shuttle back and forth within the limited terms of conventional femininity. (In Ovid at least, it’s always women who are at the precipice of crime who also break through to genuine interiority.)