The roman calendar started the year in spring as well with March (hence the names September/October/November/December are literally the 7th/8th/9th/10th month). The Greek apparently were all over the place with the year starting in summer/fall/winter depending on region/polis.

Up until 1751 the legal year (fiscal year) in England (and its North American colonies) started at 25 March (Lady Day).

One origin story for April Fool's suggests that it's to do with getting the year wrong.

My understanding was that new years was always January, 1st, but that the year number changed on Lady Day

The UK's tax year still reflects this, starting from April 6, which both accounts (pun intended) for the year changing on Lady Day, and the transition from Julian to Gregorian calendars.

I'd heard this was because July and August (for Julius and Augustus) were added later to the calendar and originally there were only 10 months that started with January.

I believe those were merely renamed, and the original year was 10 months from March to December, with a nondescript “winter” period between December and March.