Interesting question. I think that with some pieces it would be relatively obvious (>90% success rate), with some others it would be difficult but possible (>50%), and with others it would be essentially impossible (no better than chance).
The first type would be any so-called “cyclic symphonies” that make use of some themes or other musical ideas that are present in multiple movements. If I were comparing the two symphonies, and I noticed themes from the second and third movements of #1 appearing in the finale of #2, I’d be pretty sure that the movements had been shuffled. Of course, it isn’t entirely impossible that a composer would reuse themes from one symphony in another, but it isn’t very likely.
The earlier we’re talking about, the more modular these things were and the more like the 3rd type in my first paragraph. In the pre-Haydn symphonies of Wagenseil and Sammartini and such, there’s no indication that they thought of the pieces as particularly unified works. A symphony was more like a short story collection with the movements as individual stories that are only vaguely related to each other. This is still the case in early Haydn and Mozart, and still mostly the case in Beethoven’s first two symphonies. But in the later symphonies of these composers (The “London” symphonies of Haydn, the 35–41 symphonies of Mozart [not 37] and 3–9 of Beethoven) you can already hear a shift toward a more novelistic view. The movements are now more like chapters of a larger work.
So the second type of symphony might be the non-cyclic examples of some of those and most symphonists of the Romantic era. A heavier first movement might demand an even lighter minuet/scherzo (though, then again, it might demand a continuation of the tension…). The more powerful, dark or oppressive events are early in the symphony, the more likely that the slow movement would have to respond elegiacally to that tumult. I don’t think I’d have a high rate of success, but I think I’d at least do better than chance at making those determinations if we could run enough tests. EDIT TO ADD: A good example might be Amy Beach. Unfortunately, she only wrote the one symphony, but if she wrote a second that was also in E minor, I suspect it would be easy to tell if someone shuffled the movements up. This is because her symphony—although I’m pretty sure it isn’t cyclic—has an overriding theme of using Gaelic and Gaelic-esque melodies. Unless she did the same theme again, which doesn’t seem likely, it would be pretty easy to separate the Gaelic Symphony in E minor movements from the hypothetical Second Symphony in E minor movements.
Of course, your question specified that both symphonies would be in A minor; but if one began in A minor and the other began in A major, the task would probably be easier. However, many symphonies begin in minor but achieve major by the end of the last movement (Beethoven 5 and 9 are the classic examples), so even that could get a little confused. If any movements are attacca—meaning you go directly from one movement to the next like the move from III to IV in Beethoven 5)—that would be a dead giveaway.
Post-Beethoven there’s an almost universal sense amongst the famous symphonists that every symphony should be a singular statement. Cyclical symphonies become more and more common, non-standard movement structures and key relationships become the norm, and, eventually, any of the standard signposts from the Classical era are largely gone. Ironically, it actually becomes easier and easier to tell which movements belong to which symphony even as the traditional signposts dissolved. In extreme cases, such as the vast majority of Allan Pettersson’s 15 symphonies and almost all of Kancheli’s 7, they aren’t even divided into movements at all. Even the earlier Sibelius 7 and Barber 1 are single-movement symphonies, although both pieces are clearly a mashing together of the traditional four movements.