The month known as Leagueheim is the exception to nearly every rule in the Fraternity of Order's calendar - mainly because they had nothing to do with its creation. Each year, starting on a different day every time, there is a period of three weeks when the weather of Sigil drastically changes - the clouds and pollution clear up, it doesn't rain, and the air actually gets warm. (This always occurs at the same time that the Outlands experiences "The Veiled Month" of its calendar system.) While the Guvners simply tried to ignore this occurrence as just another set of days that need no special significance in their time-keeping system, the philosophical weather clears up right along with Sigil's. During this span of days, whatever month (or two) was influencing the Cage is canceled and replaced by Leagueheim. Over the centuries, these three weeks have come to be seen as a fourteenth month in the cage. And as such, it has come to be adopted by the Free League.
There is no official event that could be considered a Factol's Day for either faction or the month. However, this doesn't mean that the beginning of the month goes unmarked. On the third day of Leagueheim, the Indeps begin an informal event they commonly refer to as The Gathering. During the next eight days of the month, members of the Free League from across the planes come to Sigil and collect in the Great Bazaar. There, they meet one another, exchange information, tell stories, and generally enjoy themselves and one another's company. Part of the Free League's creed is to always have enough information to keep your options open and make an informed opinion; the Gathering is the most visible portion of the faction's famed information network.
For the rest of Leagueheim, the Indeps take different approaches to their adopted month. Members of the Free League mostly downplay the significance of the month, just as they do with their status as a faction. At most, the Indeps will be on their best behavior - aware that the Cage is watching and making sure that they lead by example. Some members will make a point of becoming touts for the month, catching the Clueless and warning them of the dangers of falling in with the factions before they've got all the facts. Others will attempt to engage members of the other factions in reasoned discourse and try to show them their errors in being too strongly committed to one point of view.
Leagueheim's influence on the rest of the Cage is different from those of the other months in the Philosophical Year - it doesn't have any. See, the atmospheric effects of the month (or months) that these three weeks override are completely negated, leaving the residents of Sigil free to think and act completely on their own. It is the one time during the entire year that everyone - from the most Clueless to the most rabid factol - is on equal footing with regards as to knowing what their personal belief system is. It is in this air of intellectual freedom that Sigil is best able to show its true, cosmopolitan character. It seems that more travelers come through the Cage's portals during Leagueheim, bringing a wealth of new chant and ideas with them. While fights between those of conflicting values and beliefs occur regularly, it is an honest exchange of hostilities such as would be found elsewhere on the planes. The ratio of beings in the City of Doors tips in favor of the planeborn, as they come to Sigil to teach mortals that there are other philosophies to live by - those of a moral basis rather than an intellectual one. Ironically, after this month of mental liberation is over, all of the factions see a sharp increase in new members.
Even though the timing of Leagueheim changes from year to year, a number of regular events still take place during this month. All deal with the theme of freedom in one form or another. One of these is the sharp increase in the number of thefts in Sigil. Others include parades or festivals which commemorate some great event (usually a battle) which resulted in the liberation of one group of people or another. Also, there are powers who find the roving nature of the month to their liking. Obviously, the gods of Travel as well those of crafts or professions which involve movement have holy days during this time. Powers whose portfolios cover the realms of Stealth or Thieves also enjoy Leagueheim because it can strike at any time to relieve the other months of their influence.