Japan's rainy season, or tsuyu, generally happens between May and July. Japanese are very particular about getting rained on, and umbrellas go up at the slightest hint of droplets in the air. The past three or four days have also been quite gusty, meaning the sight of umbrellas - especially the 500 yen, convenience store, variety - turned inside out is a common one. There is even a special word for the phenomenon: ochako ni naru, or ochako ni sareru.
An ochako is a small sake cup, the two characters for the word being "boar" and "mouth" 猪口 - a combination that, nevertheless, is believed to be substitute characters used phonetically, not etymologically. (However, it is usually written using only hiragana.) The word is also used to mean "cup-shaped."
Thus, having your umbrella turned inside out is to have it made cup-shaped: the typically small circle of a base, under which you are haplessly standing, flaring out upwards on now ramrod-straight ribs towards a large-diameter rim.
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