True enough, my mistake! I was only looking at the Roman time period and didn't realize that the calendar shifted back to starting in March during the Christian era. That's really interesting to learn!
My main point is still the same: Caesar's calendar started in January and he was following the order that was current in his day. We do not know when the switch from March to January occurred. There are multiple stories about the switch (the supposed dates range from 750 BC to 153 BC), and there are legends of a 10-month calendar that didn't include January and February at all, but there are no extant contemporary records describing the switch.
The legends about the 10 month calendar are almost certainly a very late invention, which appeared as an attempt to explain why e.g. December is the 12th month.
It is likely that at its conception all the months of the Roman calendar were numbered, but some time later the months January to June were given names corresponding to important religious festivals, which were scheduled during those months.
> but there are no extant contemporary records describing the switch.
We have something of a philosophical disagreement; I would argue that the names of the months themselves constitute a contemporary record describing the switch. It doesn't tell us when the switch occurred, but it tells us very clearly that the switch occurred.
It is true that the record might be accidentally or deliberately falsified; perhaps the months were originally named after unrelated concepts and the names got revised into their numerical forms over time. But a formal history suffers from exactly the same problem, and it still gets to be called a "record"; I don't see why we would consider the fact that our ninth month is named "Month Seven" not to be a record of a shift in the beginning of the year [or rather, in the beginning of the sequence of months].
My main point is still the same: Caesar's calendar started in January and he was following the order that was current in his day. We do not know when the switch from March to January occurred. There are multiple stories about the switch (the supposed dates range from 750 BC to 153 BC), and there are legends of a 10-month calendar that didn't include January and February at all, but there are no extant contemporary records describing the switch.