Is it fair for a professor to grade us on the possession of past papers?

If this was not mentioned in the syllabus (or at least announced prior to the papers being returned), this is totally inappropriate. While it may be foolish for students not to keep their graded work (at least until the end of the term), if it was not established as a requirement before the students got their papers back, it cannot reasonably be made a retroactive requirement.

You should bring this problem to the attention of the instructor. You do not need to reveal whether you actually have all your past papers when you bring this up, lest the instructor try to wave your concerns away if you say that you actually do have all the papers and are just complaining as a matter of principle. I would also suggest having multiple students come to her together to make the same objection; that may make it more likely that she will relent. However, if the instructor does not agree to make a change, you should take this up with someone higher up. That could mean the department chair (or another departmental official, such as the director of undergraduate or graduate studies) or a student ombudsman, if one is available.

I'll first say that if the Professor needed a copy of the graded assignments, she could have and should have just made the copy when grading, and kept it. This sounds like an arbitrary "trap" for people who are a little absent-minded, or mere arbitrariness.

First order of business IMHO is talking to the Professor, preferably out-of-class. Explain that you have not kept copies of all your work, and that you were not aware (regardless of who is to blame for it) that you had to do so - and you would like to ask for an alternative w.r.t. grading.

Considering what you've said, this is likely to fail. While waiting for the above meeting to occur, also talk to: 1. Your fellow students in this class. 2. Your student union representative (assuming you have one; otherwise - higher-ups in the student union). If they are willing, consider planning for some sort of collective action or intervention which would force the Professor's hand in case she doesn't agree to flex her positions. It's much easier to avert inappropriate grading before rather than after the fact. What could such an action be:

• Most or all students simply fail to bring any graded assignments as requested. They collectively agree to say they haven't kept copies of them. The Professor would not be able to just fail the whole class (or rather - is unlikely to do so and would be unable to defend the action) and would likely cave either at that point or earlier.
• Student union starts some sort of publicity campaign against this Professor's policy.
• Student union petitions for disciplinary procedures against the Professor (in some universities there's a legal right to start such formal procedures; in some you have to beg).

It's not that I think that these things should be done, but they are not impossible nor inconceivable and should be made possible as fallback in case things turn sour. Regardless of whether such action is agreed upon and readied for execution, either a student union rep or a group of students, and even better - both, approach the Professor collectively to try to change her mind.

But let's parallelize some more. Next step up after the personal meeting with the Professor is the Professor-in-charge for the course (in case it's a different person); then, next up - the person in charge of (undergraduate?) studies in your department. Again, student union rep + many students is the best combination from your side to meet with people further up the chain of authority.

Good luck!

PS - @Buzz's answer also makes some good points - especially about not saying which assignments you have or don't have.

To make a somewhat contrarian answer with a somewhat obvious point: is the teacher's request actually causing problems to someone in the class?

If it does, then I believe it is completely reasonable to complain, and also to side with the people affected even if you do not have the problem yourself (e.g., you have your past assignments).

However, this isn't obvious to me because the question doesn't say there is evidence for it. So, if it so happens that everyone has all their assignments at hand (or that people who didn't were not actually penalized for it, e.g., because the teacher realized on her own that it was a bad idea), then I wouldn't think it's a good idea to complain just out of principle because it could have put someone at a disadvantage or because the policy was bad. This could be given as feedback to the teacher once the class is over (e.g., when evaluating the class if this is done) but I wouldn't appeal to challenge a rule if it didn't actually affect someone.

In summary I'm just saying that it's OK to complain about an actual injustice but I wouldn't complain as vocally about the fact that an injustice could have occurred if there is no evidence that it did.