Is there a standard ordering for the question mark and the exclamation mark used together?

I think if you ask the experts who would claim that they know what the “correct” way to punctuate something is, they would tell you that a sentence may only have one terminal punctuation mark—that is to say, neither “?!” nor “!?” is correct.

So, no matter what order you use, you’ll never please those people. The Corpus of Contemporary American English has 3742 examples of “?!” and 1197 examples of “!?”. Clearly both orderings enjoy substantial usage, although it does appear that “?!” enjoys a majority of usage, probably because most sentences that get the double punctuation treatment are syntactically questions that have an exclamation point added for emphasis.

Edit: I checked in the British National Corpus, and it has 224 instances of “?!” and 121 instances of “!?”.

I guess it depends on what you want to say. In my own usage, '?!' would generally be more frequent, expressing astonishment at a particular question, kind of "WTF?!": You ask a question and then use the exclamation mark to stress its unusuality.

On the other hand, '!?' seems much rarer, both in my own usage and what I have observed. I would say it meant something extraordinary (that you stress with the '!') but also insecurity (hence it being followed by the '?') about your assessment of it being extraordinary.

So, they do express different meanings, but '?!' would be a more common one than '!?'.

Use the interrobang (quesclamation mark)‽

(I largely kid, of course. This punctuation mark is hardly in common use - though it's perhaps acceptable in various forms of media/advertising.)

In all seriousness, it is strictly only legal to use a single punctuation mark at the end of a sentence/phrase.

I like ?! and think interrobang is ugly, but that's just my personal aesthetic. As others have said, there doesn't seem to be a strict rule. Pick the one you like.

We could assume the ordering conveys information, i.e. whichever one comes first is the dominate one. So ?! would be a question asked excitedly, and !? would be an exclaimed question. But that's just us making things up. Though who knows? Maybe it'll catch on. That's my proposal for the rule.

I'm sure specific style guides and domain-specific grammars (such as chess notation, as Wikipedia notes) have more explicit meanings as well. You could probably find one to justify whatever you want.

I know you say they are irrelevant, but I would never use "?!" or "!?" but always use "?!?" or "!?!" - except in formal writing where I would use neither. I would consider the first punctuation mark to be the most important.