There's more context in the wiki article (emphasis is mine):
Traditionally, the original Roman calendar consisted of 10 months totaling 304 days, winter being considered a month-less period. Around 713 BC, the semi-mythical successor of Romulus, King Numa Pompilius, is supposed to have added the months of January and February, so that the calendar covered a standard lunar year (354 days). Although March was originally the first month in the old Roman calendar, January became the first month of the calendar year either under Numa or under the Decemvirs about 450 BC (Roman writers differ).
So, Numa introduced two new months for winter. He could have added them to the end of the year, or the beginning. It is possible it just didn't matter to the Romans.
It should be noted that even after the introduction of the two months, elected Roman officials didn't take office until March long after even the Decemviri - so the fact that January and February were listed before March on some Roman documents (and eventually adopted into the medieval calendars) doesn't signify that the Romans moved their New Year all that much. In fact, the 1st of January only seems to have become the New Year around the 16th Century (previously having been observed at Easter or Christmas in various places and times).