This perhaps indicates that one of the sides is meant for one season and the other for another season. Probably, therefore, there were two seasons and not one. This is a reasonable way of seeing it for the periods of the moon also have twos in it, Waxing and Waning Moon (and 29 and 30).

Their Sacred Year could have had two parts, one covering 4 * 31 = 124 days and the other 5 * 30 = 150 days. Nine months make me think about the time it takes for a baby to be born after conception.

Which of our modern months could they have meant? They lived north of the equator and that suggests the period from vernal to autumn equinox. But that period covers only 6 months. I think they also included the months of autumn and regarded winter as outside their sacred year.