There is a (confusing) variety of hats associated with the Kagyu school of Tibetan buddhism.
Seen from the front, the above headdresses form a tall and impressive crown-like volume.
Whereas this one has peaked flaps on each side.
And this one looks half unfolded on the sides and at the front.
15th century, Tibet, Jonang lama, gilt copper alloy, private collection, photo by Bonhams.
The hat worn by Jonang lamas is like the Indian pandita hat but yellow. This one is richly incised all over with a floral pattern that matches the cloak.
Derived from the Indian pandita hat, the standard Sakya hat is red.
The red pandita hat is often worn like a flat cap by Tibetan scholars, and was especially associated with translators. This lama holds a long-life vase in his left hand. The rim of the base supporting the lotus on which he is seated is decorated with a very ornate chased floral pattern.
This Gelug hat is quite similar to the yellow Jonang hat (The Jonang school was eventually absorbed by the Gelug school).
This type of hats, with the sides upright and considerably taller than the front, topped with a lotus bud, is worn by hierarchs such as karmapas (black with visvajra at the front), shamarpas and situpas (red with a triple gem at the front).