The school takes its name from the (転害門), the principal western gate of Todai- in Nara, outside which a lineage of smiths resided and forged swords in close association with the great temple. The founder is traditionally held to be Kanenaga, whose period of activity is placed around the Shoo era (1288--1293) of the late period, though the has noted that "from the stylistic character of extant works one also senses an earlier date than Shoo." From Kanenaga's circle emerged his son Kanetsugu, whose earliest dated works — bearing inscriptions from the Bunpo and Gentoku eras — constitute some of the oldest documented pieces from the school, and Kanekiyo, transmitted as either the son or student of the second-generation Kanenaga, whose dated Karyaku 4 (1329) provides a critical chronological anchor. The school flourished through the period and into the period, and among the five principal traditions of the , the group achieved the greatest scale of production. By the period it appears to have absorbed the other Yamato schools, continuing essentially as the sole surviving lineage through the later generations of Kanemasa, Kanetoshi, and Kanetsada, collectively referred to as Sue-.
The forging is characteristically with a tendency toward standing grain that "in places flows and becomes -inclined" — a structural signature shared across the Yamato schools yet expressed here with particular refinement. Thick forms throughout, and abundant enliven the surface, with standing out toward the in many examples. The is fundamentally , sometimes carrying a slight -like feeling, with , , or small occasionally mixed in. The diagnostic activity occurs along the , where , , , and uchi-noke appear with regularity — a constellation of edge phenomena the identifies as distinctly Yamato in character. Fine and run through the temper, and the characteristically displays vigorous , often achieving a -like conclusion. The is consistently high and the wide, reinforcing the structural identity of the school. The single most defining hallmark — and the criterion by which the distinguishes works by Kanenaga himself from broader attributions — is the exceptional strength and brilliance of the : "rounded, beautifully lustrous" particles that sparkle throughout both and , producing a that is "extraordinarily bright and clear." Among signed works of the Yamato Five Schools, Kanenaga's blades display the most strongly -laden tempering, at a level the has judged "can rival the finest masters of ." Later generations show a natural evolution: the Sue- period broadens the tempering vocabulary with -based - and intermixed, lending certain works a -like tendency that closely resembles Sengo Muramasa.
The school's legacy within the Yamato tradition is one of singular preeminence. Kanenaga is consistently evaluated as the foremost smith of the Yamato Five Schools, and two broad categories within his oeuvre are recognized: calm, subdued works with lower tempering, and more animated examples with higher in which the differs between and , as exemplified by the celebrated "Kotekashiwa Kanenaga." Kanetsugu's signed, pieces with date inscriptions from the Bunpo and Gentoku eras are deemed "of the utmost value as documentary material," while Kanekiyo's production across as many as seven successive generations offers an unusually complete record of the school's evolution over more than two centuries. Kanemasa and Kanetoshi sustained the tradition through the period with workmanship that the terms "unmistakably neat and careful," and Kanetsada's rare long inscription including "Nanto-ju" provides evidence of the school's continued vitality into the final phase of production. Distinguished provenance attests to the enduring esteem in which these blades have been held: examples bear gold-inlay attributions by successive generations of the family and document ownership by Honda Tadatoki, the Sakai family of Himeji, the Mizuno family, and the Satake retainers. Across the full body of examined works, the 's evaluative language converges on a single axis: the luminosity of in both and , and a that is bright and clear — qualities that define the school as the standard-bearer of the Yamato tradition at its highest level of refinement.