The history of the Roman calendar is somewhat unclear. The Julian reform is well-attested, as is the immediately pre-Julian calendar. This latter calendar is 12 months starting with January with day lengths of 29/28/31/29/31/29/31/29/29/31/29/29, plus an occasional intercalary month of 27 days whose insertion caused February to shrink to 23 days.

The Romans themselves maintained that before the Republican calendar, the year consisted of 10 months of alternating 30/31 days, with the winter months (of what would be January and February) consisting of a vague collection of days omitting a fixed month. But this explanation is dated essentially to people writing "what we used a millennium ago," and there's no attestation of this calendar much closer to its usage period.

In any case, it's definitive that the year used to start in March. When and why it transitioned to January is unclear. For what it's worth, even after most of Western Europe transitioned to using a Julian calendar, the starting date was quite inconsistent until very late. (England only switched in the 18th century, at the same time it adopted Gregorian calendar).

I had believed something similar. They are definitely remnants of a 10 month calendar, but July and August were existing months (Quintilis and Sextilis, respectively) renamed for the emperors. January and February appear to be the additions [0]. There is quite a bit of interesting history in the rest of that article about how the months have changed.

I can't remember if I'd read about a past society that did this, or it was an idea someone had proposed for a future one, but I love the idea of a 360 day year (12 months, 30 days each), with a 5 or 6 (leap year) day festival at the end commemorating the New Year. It would change birthdays for everyone born after January 30, which seems like it'd cause enough outrage to prevent it, but it's a fun thought!

[0] https://en.wikipedia.org/wiki/Roman_calendar#Legendary_10-mo...