Question: Let's say user A follows user B, and B follows A. I want to know the exact date A started following B and viceversa.
Is this information stored on twitter? Can I retrieve it using the API?

To clear out: The point of this question is finding a way to know who followed who first.
(I'm assuming both A and B deleted the notification e-mails)

Answer: 
No Ignacio, you can't. You just can know who follows who but not the date the follow started.

Looking at the API, there's is no way, there are two calls to get the followers:

User Methods/statuses/followers

and

Social Graph Methods/followers/ids 

Neither of them returns dates or even a serial that would let you see who started following first. Really, **there's no indication that twitter is internally storing this information**, neither in the API nor Twitter's web interface.
This is a very old question, but perhaps some might be interested to know that while you cannot get the date at which someone started following, you can at least infer an "earliest possible following date" from the fact that the list of followers is ordered according to date, and the fact that follower objects come with a `created_at` timestamp. 

Here's a Python function for calculating an "earliest possible following date": <https://github.com/BernhardClemm/twitter-follow-dates>