Both the Walrussian and Harlovian comments are right. The turban around a tarboosh is worn by the ulema = the learned men in islamic religion and justice.
The custom to wear it around a helmet is as obvious as the use of the tin hat in modern times.
Turbans made to last are regularly the most complicatedly folded, so you spare a lot of work every time you put it on. There you need a cap, too.
PUGGRY, PUGGERIE s. Hind. pagrî, 'a turban'. The term being often used in colloquial for a scrf of cotton or silk wound round the hat in turban-form, to protect the head from the sun, both the thing and the name have of later years made their way to England, and may be seen in London shop windows.
Hobson-Jobson : a glossary of colloquial Anglö-Indian words and phrases ... / by Henry Yule and A.C. Burnell. - First publ. in 1886, 2. ed. 1903. - New ed. - Sittingbourne : Linguasia, 1994

Wilfried

Special signal for Larry: Thanks a lot! Wilfried