Mujitsu and Tairaku's Shakuhachi BBQ

There could be something to that.  I've never worn a tengai so cannot answer that question.  There is some information on tengai in Sanford's article "Shakuhachi Zen: The Fuke-shu and Komuso" in Monumenta Nipponica 32, 1977.  Sanford says that up until about 1700 the tengai was no more than a fairly shallow sedge hat and from about 1720 on it was transformed into its more familiar form, "a sort of inverted reed basket which covered the wearer's entire face."  According to the rules of the Fuke sect, the komuso monk was never supposed to remove his tengai except within the precints of a Fuke temple and "if caught outside in a rainstorm, he was not to open an umbrella over the tengai."  He also notes that unauthorized use of the tengai was a punishible offence, but that enforcement of that rule grew lax over time and that samurai traveling incognito took to wearing the tengai.  Nothing about blowing in the wind though...  I was trying to see if Kaempher mentioned komuso in his Tokugawa era diary but the only thing close I could find was a description of his visit to Fushimi, south of Miyako (Kyoto) where he writes "We also saw various strangely clad beggars: some dressed up in mad costumes, some walking on iron stilts, and others with deep buckets on their heads with green trees.  Still others were singing, whistling, playing the flute, or striking chimes" (Kaempher's Japan: Tokugawa Culture Observed, Book 5, p. 318); the deep bucket hat guys could have been something else altogether...

You might want to take a look at the photos of prints showing different types of street musicians in the Edo Period in their various hats (as well as photos and drawings of musicians with their hats from other time periods) in William P. Malm's Traditional Japanese Music and Musical Instruments, the one source on Japanese traditional music which most university libraries are likely to have.  Specifically, plates 19 and 20 in the most recent edition (2000) of the book, which was originally published in 1959, under the title Japanese Music and Musical Instruments.