As it turns out, there are a lot of opportunities for you to switch between which side of the Civil War you are helping rather than stick with a side and run with it. Let me show you what I mean:
– At the beginning of the game, escape with Ralof.
– Instead of taking his advice of signing up with the Stormcloaks, go to Solitude and do the Joining the Legion quest up until the point where you have to take the oath.
– Instead of taking the Legion's oath, go to Windhelm and complete Joining the Stormcloaks, taking their oath instead.
– Go with Galmar to Korvanjund and retrieve the Jagged Crown, but take it to the Legion instead.
– During Season Unending, create a ceasefire agreement that favors the Stormcloaks. Fun fact: According to UESP, Bethesda considered options for this quest that would have you kicked out of your chosen army if the negotiated truce favored the other side. From there, you could switch to the other side. Of course, this didn't make it into the final version of the game, but it certainly would've been cool if it did.
– All the while, continually switch between which side you help during the relevant world interactions. For example, if you come across some Imperial soldiers with a Stormcloak prisoner, you could fight to free him, but then the next time you come across a Stormcloak patrol, you could kill them all.
Of course, you could accomplish the same thing by escaping with Hadvar at the beginning of the game and flip-flopping from there. Most, if not all the stuff I covered is well-known by now, or at least is information that can be easily found on UESP, but this shows how you could do this all in one playthrough. Does this let you accomplish anything useful? Not really, but it could make for some interesting roleplay (like if you were playing a Thalmor agent trying to weaken both sides, for example), and I think it's interesting enough to warrant sharing here.