The Sakya combines two things that usually pull apart: a tantric transmission held inside one family, and a tradition of sharp public philosophical argument.
The path with its fruit
Lamdre, path and fruit, is the school's central teaching, traced to the Indian adept Virupa and organised around the Hevajra tantra. Its distinctive claim is that the path and the goal are not two stages but one thing seen from inside and from outside, so the fruit is not added at the end. The teaching was transmitted privately for generations and only committed to a public commentarial literature later.
Sakya Pandita
The most consequential figure of the school, in the thirteenth century, wrote on logic, on poetics, and on what he took to be corruptions in the tantric practice of his contemporaries. His Treasury of Reasoning imported the Indian epistemological tradition wholesale into Tibetan scholastic training, and his polemic against what he called the Chinese style of sudden meditation set the terms of a debate that ran for centuries.
Held in one family
Authority descends in the Khon family by blood rather than by recognised rebirth, which is the opposite of the Kagyu solution to succession. The headship alternates between two palace lines. That arrangement kept the school's leadership out of the recognition disputes that have troubled the incarnation-based schools, at the cost of tying its continuity to the survival of a single lineage.