I’ve seen this used in the context of statements like “x is less than, equal to, or greater than y if and only if w is less than, equal to, or greater than z”. Which might be written “x ⪋ y if and only if w ⪋ z”.

When is x not less than, equal to, nor greater than y ? Is it like, x is complex and y isn't?

The original text is meant to mean a triple of statements:

x < y if and only if w < z

x = y if and only if w = z

x > y if and only if w > z

This can be tiresome (and obviously confusing) if you need to keep using it in many places, hence some authors define the odd symbol to start with, and then use that in many places.

“If and only if” also has a shorthand in “iff”.

Surreal games[1] are probably my favorite example. But any partially-ordered set without total order can be an example.

[1] https://en.wikipedia.org/wiki/Surreal_number#Games

For partial orderings, we can have neither a ≤ b nor b ≤ a. Perhaps it's that?