The Imperial Calendar is based on the Neo-Babylonian calendar (which was actually a thing). It’s “Lunisolar”, which is to say the months are based on lunar cycles (the first night of every month is the young crescent moon after the new moon, and the last is the night of the new moon), but the years are based on solar cycles (the first day of every year is the Spring Equinox).
Practically, this means that New Year’s day doesn’t fall on the same day every year (like, “January first”), but wanders around a bit on the calendar. Because there are 354 days in twelve months of new moons, and 365 (and change) days between spring equinoxes, new years days moves around by 11 or 12 days every year.
This is where “intercalary months” come in. Bear with me. Intercalary months means that every so many years, you add an extra month into the calendar, to make up for those missing eleven and change days. And then New Year’s day falls back on the first day of the first month that year, and then you start losing days again.
Anyway, the Imperial calendar has a 19 year intercalary cycle. In six of those years, you get an extra month. And because fractions, in one of those years, you get a second extra month.
Practically, it looks like this:
(That three month range in the “approximate equivalent” column is because of the previously mentioned “months wander around on the calendar” thing that gets resolved every 19 years.)