I believe it's proper etiquette for the loser to request the review?
Does anyone know the answer to this?
If I'm unintentionally being rude -- that might explain the rude replies! As I see it, I often review my games regardless of the outcome. So, when I have time for a review, I always invite my opponent -- out of both politeness and selfishness. Even if I won, I'm sure that my opponent had ideas that did not occur to me.
And if this is the norm, this seems odd to me -- reviewing your game is generally useful. So, why should there be an asymmetry in who gets to extend the offer? In fact, my initial thought on this was that excluding my opponent from a review would be rude.
In an ideal world, I'd agree with you, but in reality I'd agree with swannod. If I just lost a game to someone who has the same rank as me and this person asked me to review, it really feels like this person is saying to me "Let's go over your mistakes". Sure that's probably not the case, but I just lost a game, so I'm not in the best mood and I'm probably more emotional than rational at that point in time (i.e., if my opponent PM'd me a day later asking me to review our game, I'd be much more inclined to agree). And since my opponent is the same rank as me, as shallow as it sounds, I'd refuse to think that my opponent could come up with anything in a review that would be insightful to me. After all, my opponent is the same rank as me and I could just as easily come up with anything he or she comes up with! But in an ideal world, even the opponent that's 6 stones weaker than you who just won you in a 6-stone game could have ideas that didn't occur to you, right? It just comes down to pride.
Well, I for one am generally grateful when a better player (i.e., someone who just slaughtered me) offers a review. I rarely feel like it's my privilege to demand/request a review from somebody after a casual game.
So, in my opinion, there's nothing at all wrong with asking the other player if they want to review a game. If the other player has a fragile ego after losing, and would rather sulk than take the opportunity to improve, that's their problem.