The first house is a pivot, one of the four kentra that the Latin tradition calls angular, and in the Hellenistic scheme it is the Horoskopos, the hour-marker. Every one of the twelve places is judged in relation to it.
On the cusp itself the house division systems come closer to agreement here than anywhere else, and they still do not agree. Equal, Porphyry, Alcabitius, Campanus, Regiomontanus, Placidus and Koch all begin the first house exactly at the ascendant degree. Whole-sign houses do not: there the first house is the entire rising sign, so the ascendant sits somewhere inside the house rather than at its head, and a planet in the same sign as the ascendant but a few degrees earlier is in the first house under whole-sign and in the twelfth under every quadrant system.
That single divergence is the most consequential in practice, because the twelfth is among the difficult places and the first is the most personal, and the same planet is read very differently in each.
The house also fixes the classical geometry of the chart. A place is said to be configured to the Horoskopos if it stands to it as a sextile, square, trine or opposition, counting by whole signs. The third, fourth, fifth, seventh, ninth and eleventh are configured. The second, sixth, eighth and twelfth are not, and are said to be in aversion: unable to see the rising sign at all. Those four are precisely the four places the tradition treats as unfortunate, which is not a coincidence but the reasoning behind the judgement.