Well, with the word "can" the sentence basically means "he could stand up any moment now and turn the world upside down", whereas using "could" would make it look like a possibility "yes, if he was to try, he could in fact turn the world upside down". But I don't think it's wrong to use the word "can"?.. It conveys a different feeling compared to "could". "Can" makes it feel more in the now. If that's the effect you're going for, I think it's perfectly correct.

It's consistent with the uses of "he is" which are also in the present, which makes the narrative sound more immediate, even if the events were actually in the past. (This usage is sometimes called the "historical present".) It would be perfectly OK to put those verbs all in the past, too, i.e., "he was" and "he could". But to use "he could" with "he is" suggests that his ability to invert the world has now come to an end.

-> He is an extraordinary genius who 'can' turn the world upside down. - the use of "can" make the sentence appears as a fact which means that "no matter when or where the person is, he or she can turn the world upside down" -> He is an extraordinary genius who 'could' turn the world upside down. - Also, the sentence appears as a fact but the only difference is that it's in the past, (the fact is already done); and the person may not be able to turn the world upside down.