Ok. No, it is not what you are probably thinking. Hentaigana has nothing to do with perverted stuff so stereotypical in popular culture. This hentai we are talking about here pertains to this thing called 変体, or variant forms, and that hentaigana, or 変体仮名, basically means the historical variants of the currently used hiragana script.
If you browse through rather historical writings, calligraphies, or perhaps traditional Japanese art forms, you might come across some characters which do not seem to resemble any kind of character you may have learned in Japanese classes. You see, the hiragana we use today typically follow a one-to-one correspondence between character and consonant-vowel sequence. But this was historically not the case.
Before the Meiji Restoration, there were several interchangeable hiragana that could be used to represent a single consonant-vowel sequence. As the 1900 script reform occurred, only one of these characters per sequence was deemed hiragana, and the rest, hentaigana.
It is widely known that hiragana derives from this thing called man’yōgana, where kanji characters were written in cursive, smoothing out some strokes normally seen in kanji. You may have also seen some charts showing which kanji characters the hiragana script we use today derive from. While rather informative, it misses out on the larger family of hentaigana‘s evolution and history, where more kanji-derived or man’yōgana-derived forms were used. For instance, the hiragana あ (a) is derived from the kanji 安, but its hentaigana counterparts could also derive from characters like 阿, 愛, and 悪, all with their own connotations and contextual meanings.
It is proposed that the use of hentaigana in historical texts, calligraphy, and art forms allowed the use of certain characters to bring out a certain meaning or connotation in a text, which perpetuated the privilege some people had when Japanese was still written in man’yōgana, and such education was limited. With the 1900 reforms, however, learning how to write Japanese was made more accessible, as a one-to-one correspondence between character and consonant-vowel sequence was implemented, compared to the many-to-one correspondence used in antiquity.
While hentaigana do not seem to render properly in computers, there is a site which holds many of the commonly derived hentaigana from various historical texts, for everyone interested to admire. You can find them here:
What is interesting to note is that there were hentaigana created that represented some abbreviations of words like こと (method), とき (time), and とも (friend), although how these were derived are not shown. Nevertheless, it shows the more extended family of hiragana, in which the writing system extended the syllabary to the range of hundreds of characters.
Although many Japanese speakers and learners are unable to read hentaigana today, their pronunciation could still be deduced from context, or their use in certain traditional shop signs, or stylised writings. Hentaigana‘s use today appears to give off a more traditional atmosphere in a shop, brand, or anything in the creative sphere. While now defunct, it is amazing to see how a historical relic is still used (albeit rarely) today.