What is the meaning of the name of the intercal symbol?

The earliest mention I can find is the SGML standard (1986) which lists intcal in its additional mathematical symbols list under "binary and large operators" the definition being (in full)

<!ENTITY intcal SDATA "[intcal]"--/intercal B: intercal-->

which means the the entity ⊺ could be used to make an unspecified system specific character (with as for all characters in this standard no encoding or pictorial clue what the character should look like). Note however the section heading and the /B in the entity definition comment both imply that the intended use is as a binary operator (which would imply that using it as a funky T for matrix transpose is a mis-use)

Unicode 1.0 added INTERCALATE at U+22BA citing SGML intcal as a prior character

https://www.unicode.org/versions/Unicode1.0.0/ch06.pdf

In MathML we assigned alias ⊺ to this and defined it (as required by XML which does not have SDATA entities) to U+22BA so MathML defines

<!ENTITY intcal           "&#x022BA;" ><!--INTERCALATE -->
<!ENTITY intercal         "&#x022BA;" ><!--INTERCALATE -->

so both of names ⊺ and ⊺ work in MathML, and so in HTML since version 5, which incorporated the MathML entities.

Separately the AMS (AMS this time standing for American Mathematical Society, not the AMS in ISOAMSB which stands for ISO Additional Mathematical Symbols set B) produced the TeX AMS Fonts and assigned the character the name intercal as a binary operator with definition (in its later latex2e form)

DeclareMathSymbol{intercal}     {mathbin}{AMSa}{"7C}

The unicode math package keeps the intercal name defining it, as in MathML, to be U+22BA, again as a binary operator

UnicodeMathSymbol{"022BA}{intercal                 }{mathbin}{intercal}%

So U+22BA is intercal and looks like ⊺ which is sort of like a squashed dropped sans serif T, not to be confused with T (U+0054) which looks like T, or top (U+22A4) which looks like ⊤.

As the editor of at least some of the specifications mentioned above I can confirm that the name has just been inherited without any actual knowledge of what the character is supposed to be used for.

Wikipedia suggests "intercalate" may be used in

timekeeping

Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.

Chemistry

In chemistry, intercalation is the reversible inclusion or insertion of a molecule (or ion) into materials with layered structures.

University Admin

Intercalation, in the context of university administration, is a period when a student is allowed to officially take time away from studying for an academic degree.

But nowhere do I see any use of a symbol resembling T

Although this is an old question, it came out at the top of my search for what this symbol is about so I thought I'd add what I found - according to the Online Etymology Dictionary (https://www.etymonline.com/word/intercalate) "intercalate" came from a term referring to the insertion of a calendar date to balance the Roman calendar which was done after the Terminalia (see also the link above by David Carlisle on timekeeping).

According to Wikipedia the Terminalia (https://en.wikipedia.org/wiki/Terminalia) was a festival in honor of the god Terminus (https://en.wikipedia.org/wiki/Terminus_(god)) the god who protected boundary markers and whose name was used in Latin as the word for boundary markers.

I couldn't find anything linking this to LaTeX, unicode, or any of the other references above, so the fact that there's a T involved might be just a coincidence, but it's been two years since this was posted so I'll offer it as possibly-maybe folklore that might be interesting to the trivia-minded sorts like myself.