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

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 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 '!?'.

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.

From a more functional standpoint:

If the context is right , I will follow Occam's razor and achieve the same result but with economy.

For example, in the context of "Who do you think you are!", a single exclamation mark is not only safe and correct, but still as punchy.

I would agree that ?! isn't standard English, and I'd see it as frivolous in formal writing anyway.

Remember not to overuse exclamation marks, too.

Edit: If you still feel that adding an exclamation point for a question doesn't seem right, I feel that this work best for one-word sentences like "What!" and etc.

Putting both marks together does, to quote the Oxford Style Manual, "strike a note of almost hysteria" in serious writing.

To know which mark to use, you need to decide whether you expect an answer or not.

"What are you doing?" = you don't know what the person is doing.

"What are you doing!" = you do know what the person is doing (and probably don't like it).

The standard way of writing that combination would be “?!” (question mark followed by exclamation point), since you are first and foremost asking a question.

It is this question in which you wish to accentuate, to show surprise, anger, disbelief, etc.

The question itself is exclaimed, so it must be ? first, then punctuated ! to augment the original question.

It is possible that this practice originated in the annotation of chess games, where the two punctuation marks (! and ?) are used to identify exceptional and questionable moves respectively. In that context, !? identifies a move that looks exceptional, but has hidden flaws, and ?! identifies a move that appears questionable but contains hidden virtues.

When only one instance each of the two punctuation marks is used, I believe this inference might be picked up by most astute readers when use of two punctuation marks was at all warranted.

With the above stated, I must now ask for the reason why you are doing this? (And asking this question.) Over use of punctuation marks has the opposite of the intuitive effect, becoming mere noise that drowns out the signal contained by the actual written text. While rules in English are meant to be broken, breaking them without a clear intent in mind is simple mindlessness.