After reading this thread again, I'm thinking about a pair of names - seo-friendly is_sorted (or is_ordered) implying proximity to sort plus reduce_ok or similar that is for arbitrary checks. The implementation may be the same under the hood. There's no way to tell whether user supplied a total/partial order like gt/le or arbitrary structure check or mix of both.

I'm also thinking about incorporating compound checks suggested by 1nickt via subtests, if Test::Builder allows me to (but it should).