Onion and Garlic, a no-no?

Clearly, your flatmate was misinformed. Firstly, Italian cuisine is defined regionally. There are vast differences throughout the country, usually defined by local ingredients and historical influences. However, there are many Italian dishes...from north to south that contain both onion and garlic. It could be true that someone's specific recipe for, spinach, for example may only have onion...or only have garlic, but if you like the combination, there is no reason not to use it. There is no such general "rule" in Italian cuisine.

I agree with moscafj's general answer: there is no pan-Italian "rule" like this in Italian cuisine. It's common to mix the two in many Italian regions, and it's certainly common in other world cuisines.

On the other hand, I wouldn't dismiss this story out-of-hand or as some quirk of one crazy roommate.

I grew up near an old Italian neighbor, daughter of Italian immigrants. Her mother-in-law also lived with her, a direct Italian immigrant born around 1900 (immigrated sometime before mid-century). It was an absolute rule in their house that onion and garlic should never be combined in the same dish. And yes, they associated this with "the Old Country."

Alas, I don't know what region they were from, and they both died long ago. And I wouldn't pass on this anecdote here except I did a few quick internet searches and discovered that a few people have asked similar things on various internet forums concerning Italian cuisine. And both Mark Bittman and Gino D'Acampo have apparently passed on tales that Italian cooks have told them the same thing. (I didn't track down links to the original sources, but I have no reason to doubt these forum posts are simply making this stuff up.)

I've eaten food in Italy that clearly contained both. I know Italians who like cooking, and I've never heard them talk about this. On the other hand, I've heard this particular "rule" a couple times, and it seems only associated with Italian cuisine. I'm not really interested in trying to track down more information on it, but it sounds to me like it's at least part of cooking "lore" for some Italians, perhaps from a particular region or something.

EDIT: Just for one Italian source which seems to reference the issue, see here. After a discussion of separate uses for onion and garlic, the question of whether to use them together comes up:

Il dubbio, a questo punto, viene: è possibile utilizzare aglio e cipolla insieme? La questione divide da sempre gli appassionati di cucina.

Basically, this passage strongly implies that there are passionate cooks out there who argue about whether it's even possible to combine the two. (The link goes on to argue that it is possible, but nevertheless it references this question as if it were a common dispute.) The opening paragraph also implies there are traditional fixed rules that dictate the specific occasions when they could be properly combined.

My grandmother always told us to never mix garlic and onion. She was Calabrese. I use both in some recipes. Granny was not the best cook.

Mum was from San Sebastiano al Vesuvio, just outside Naples. She never mixed onions and garlic in her cooking. Pasta e fagioli for example she just used garlic. For Minestrone just onions. I don't remember mum putting garlic in her sauces either, she always used onions. The flavour is so unique when you use only garlic for the pasta e fagioli. Pizzas had garlic, oil and oregano on but nowadays basil seems more popular. Garlic mum used with oil and lemon juice on boiled and cooled cauliflower, carrots or green beans as a side to meat. She used garlic in braciole. I have always followed her suit, but my son likes to combine onion and garlic. I think I did hear Gino d'Acampo say once that keeping onion and garlic separate was a Southern Italy thing, but nowadays many cooks use both together. I suppose best to try both ways and see what you like best! At the end of the day it doesn't really matter.

You may hear this rule from French people, and the reason is quite simple: garlic is predominantly used in the South and onion in the North (a bit like olive oil and butter). Since traditional recipes usually have a defined geographical origin, it is quite rare to find some combining both ingredients.