For all the rejection of opinion and speculation in the latter part of this thread, there is quite a bit of it happening regardless. Speculation is a fact of life for this game, let's not pretend otherwise, eh?
It's made clear that it's very soon after Torygg has been killed. The player character is caught amid members of an armed insurrection. Your character is rounded up and hauled off. Yeah, it's true that you are not on the list. If Bethesda wanted to make the Empire and Legion look completely and irredeemably heartless and bloodthirsty, we would NOT have seen a very sympathetic Hadvar who asked the captain next to him what to do about a hapless soul not on the list. It's made clear that at that moment, General Tullius is wholly preoccupied with the most wanted man in Skyrim. His back is to everything that's going on with the careless captain and Hadvar. Is this a fault of his? Perhaps, perhaps not. Fact is that the player character was going to be executed solely on the whims of one irritable and reckless officer - you were NOT on the Legion's list. As was also arguably the case with Alduin's appearance, it was a chance affair - you were the near-victim of an officer who appeared to just really not give two bones about you or any sort of code. It's easy to conflate that with the Legion in general, but it's unfair to do so and I will continue to maintain that whenever that particular argument surfaces. If anything, I say the dismissive captain is a necessary facet of game mechanics.
As for speculation and opinion, it's not difficult to make educated guesses about what would have happened if the tables were turned at the beginning. I can't and won't take seriously the idea, even for a moment, that the Stormcloak patrols would have acted any differently had my character been found among Imperial soldiers who just had taken Ulfric's head off. Nor would I have expected them to do so.