The flags are flying at half-mast today because of the state funeral for artist Pro Hart in Broken Hill. Traditionally, flags are flown at half mast if someone has died or to indicate some other mourning such as the Tricoleur being flown at half mast from most public places whilst under Nazi occupation.
Not only is the flag on the council chamber at half mast, but it's also upside down. This is sometimes done to express a state of mutiny which in the grand scheme of things, isn't likely because technically Mosman would still be part of NSW ergo Australia, who's flag is upside down.
Flags are sometimes flown upside down for other reasons. If a crew abandons a ship in distress to save their life, the last thing to do (if they have time to do so) is to turn the flag upside down. This means that they give up any right on the vessel or cargo and anybody who manages to rescue the ship afterwards could keep it.
Some navies in the age of sail apparently inverted the ensign of an enemy ship after they captured it, although the normal procedure was to hoist their own ensign above the enemy ensign aboard the prize. On the other hand, it would obviously be useless to invert the French ensign as either a distress signal or as a sign of capture, while the fact that some others were inverted (Spain, Libya, Portugal) would only be visible at very close range. Not to mention that there are a few flags whose inversion would simply make them someone else's flag as in the case of Ireland.
Could it be that Mosman Municipal Council is in distress or is free to capture? I would very much like to be Town Mayor through minimal force, then maybe I could get a parking permit.