Is "teenage mutant ninja turtles" grammatically correct?

Solution 1:

There are a couple of ways of analyzing the noun phrase 'Teenage Mutant Ninja Turtles'.

The first unsatisfying way is to claim that it is a one-time invention or idiom that was created rather desultorily, and as an invention (or idiom) it is correct by fiat. Even though it might be out of order according to conventional rules, it is grammatical because that ordering is the way it is created to be. This is unsatisfying because you are looking for a reason for that particular sequence and this justification works for anything.

The second way, which I think is closer to what you seek, is to analyze according to the rules of ordering adjectives. You can find this addressed quite well here at "What is the rule for adjective order?". To combine the slight variations there, the general rule in English is

article, number, judgement/attitude, size/length/height, age, color, origin, material, purpose, attributive-noun, target noun

All that is needed is to judge each of our modifiers to be these roles.

One may have more than one of each role and then anything goes (subject to vague 'sounds better' rules).

And there may be difficulty in assigning role.

• 'teenage' is pretty obviously an age
• 'mutant' is ... hm, is that an adjective or a noun? If it is an adjective it seems likely to be one of judgement (coming earlier in the order). But it feels more like a noun.
• 'ninja' is a noun, so it acts as an attributive noun here
• 'turtle' is the target noun

If you accept mutant as a noun then teenage comes first, then mutant and ninja could be in either order, followed by turtle. 'Teenage ninja mutant turtles', ignoring the prominence of the existing idiom, I think sounds fine. But the one we hear everyday sounds fine also.

So the end judgement is that the ordering is grammatical.

Of course, your inner assignment of roles to these words may well be different ad that may account for your questioning.

As a side note, newspaper headlines tend to convert numerous qualifiers into a pile-up of attributive nouns which tends to overwhelm our parsing mechanisms, but presumably they are parsable somehow to be considered 'correct' as in "Slough sausage choke baby death woman jailed". TMNT is just a large mouthful to swallow and because of that sounds 'off'.