In a video by Religion For Breakfast, around the 27.5 minute mark it is mentioned that for a significant period in Ancient Rome, August 28th was celebrated as Sol Invictus (as opposed to a date closer to our Christmas). This date, proximate to neither equinox nor solstice, at first seemed arbitrary to me, but then I wondered could 8/28 have been selected due to its falling around the center of the figure eight of the analemma? And thus the apparent point at which the sun crosses its own path in the cycle of the year? Would they have recognized and applied such a phenomena to their calendar? Apologies if this is the wrong place to ask such a question—I am new to Reddit—but I've been wondering for several months now.
A couple of clarifications. As the video points out, there were multiple festivals of Sol at any given time. And the 28 August one was not a festival of Sol Invictus: the Invictus festival is specifically a festival named in the 4th century Philocalian calendar. Sol could sometimes be called Invictus, but they're not the same thing: Sol wasn't always Invictus, Invictus wasn't always Sol.
As the video mentions, early imperial fasti put festivals of Sol on 8 and/or 9 August, and 11 December, and maybe on 28 August. From what Hijmans says (Sol: the sun in the art and religions of Rome, 588), The 28 August one is actually tentative. Unfortunately I don't have access to Degrassi's Fasti anni numani (Inscriptiones Italiae vol. 13.2, 1963), which is where the epigraphic evidence for that date is to be found. So I don't know which fasti are the source for the date, or whether the interpretation is reliable.
Hijmans, too, isn't aware of any solar significance to any of these dates, including the tentative 28 August date. Page 588:
The traditional feast days of Sol, as recorded in the early imperial fasti, ... are all dates that are unrelated to any important celestial alignment of Sol, such as the solstices and equinoxes.
That is, your suggestion appears to be a novelty. It's an interesting idea. On the one hand, it's a pretty good match, and there's no reason I can think of for dismissing the idea of tying a festival to the centre of the analemma. On the other hand, the Julian calendar in, say, the 1st century CE, ran a few days ahead of the Gregorian calendar, so Romans at that time would have placed the centre of the sun's analemma a few days into September. Also, there doesn't seem to be any corresponding significance to mid-April, when the sun passes the centre of the analemma on its return journey.
It's interesting, but given that the other festival dates are all unrelated to solar cycles, I think there'd need to be some stronger evidence for the theory. It looks likely that not much was set in stone, given that there are plenty of fasti that don't mention Sol festivals at all, and that there's not much overlap in dates between the early imperial fasti and the Philocalian calendar.
Edit: I've watched another minute or two of the video, and there are a few minor inaccuracies in how they represent things: I see that the speaker happily treats Sol and Invictus as the same thing, so it's easy to see where your misunderstanding came from. Also, the speaker mistranslates natalis, the word used for the 4th century 25 December festival: it only means 'birthday' in an etymological sense. In the Philocalian calendar it's used for any kind of celebration -- e.g. 25 January, a natalis for the arrival of the annual papyrus shipment.
(I bet they say the 25 December festival was instituted by Aurelian in 274 CE. Do they say that? Sorry, I don't have time to watch a 50 minute video just now. If I've slandered them, I apologise!)