The Discworld month roughly analogous to Roundworld's July. follows the standard pattern of having four eight-day weeks and thirty-two days.
On Roundworld, July is named in honour of the Roman leader Julius Caesar. (Just as the following August is named for the first Emperor, Augustus Caesar.)
Can we infer that in old Latatian days, there was a Grune or Grunus Caesar who was the Discworld version of our Julius? There is certainly a personal forename "Grunworth" that just might imply a survival into modern Morporkian.
"This being the time of change when the sun doth rise on the right when facing the Hub" (from a Discworld diary page).
Grune is also the god of unseasonal fruit.