The latest standard (2019) is divided into two parts: ISO 8601-1 and ISO 8601-2. Part 2 is titled "Extensions" and includes many different - but optional - ways to extend representations of dates and times. In my opinion, Part 2 represents the "leftovers" - that which was too domain specific or esoteric to warrant broad implementation.

Representations of seasons, quarters, etc. is one of many types of extensions included in Part 2.

These extensions are always used in "YYYY-MM" format (and always include the "-"). For the "MM" field, you use the numbers 21-41 inclusive. The numbers represent, in order, spring, summer, autumn, and winter, in 3 ways: independent of hemisphere (21-24), northern hemisphere specifically (25-28), southern hemisphere specifically (29-32); quarters 1-4 (33-36), quadrimesters 1-3 (37-39), and finally, semestrals 1-2 (40-41).

This, in combination with the fact that the lookup table is paywalled, makes it a great competitor for the "extremely pointlessly overengineered standards" award.

I wonder if there is any software that actually supports that.