What do you call a shirt with a very high, sometimes rolled, collar?

A shirt with a very high, sometimes rolled collar is called a turtleneck.

The garment has a close-fitting collar that rises around the neck and may fold over itself. The same style is known as a polo neck, roll-neck, or skivvy in different regions. A mock turtleneck looks similar but has a shorter collar that does not fold over.

High-neck garments have existed for centuries, but turtleneck-like clothing became especially practical among sailors, athletes, workers, and military personnel because it helped protect the neck. In the early 20th century, the style moved further into mainstream fashion. By the 1920s, polo necks were sometimes worn as a less formal alternative to a shirt and tie.

The term can describe the collar itself as well as the whole garment, and it is not limited to sweaters: turtleneck shirts, dresses, and dresses made from lightweight fabrics are also common. It should not be confused with a cowl neck, which is looser and draped, or a mandarin collar, which stands upright but does not fold over.