What punctuation should be used to indicate a set of quotes has been rearranged?

There's no need for any punctuation there. An ellipsis is to "signal the omission of a word, phrase, line, paragraph, or more from a quoted passage," in the words of The Chicago Manual of Style (§13.50). Between your closing and open quotation marks there's no omission, of course, so no ellipsis necessary, and the sentence is an acceptable English construction that doesn't need, e.g., a comma, which would in other cases be appropriate. As far as I know, and as you note, there's no punctuation mark or other indication for the rearrangement of direct quotations taken from the same source.

Thinking of this slightly differently—say, if an author were quoting two (less well-known) phrases from the same page of a single source but quoting them out of order—we wouldn't expect an author to indicate that the order of two quotations differed from the order of their appearance in the original, unless the order itself was somehow material. As long as the original's overall meaning is conserved, exact wording is enclosed in quotation marks, and attribution is given to the original author(s), the order in which one presents the pieces of a work being discussed isn't necessary to signal (unless of course it's part of the argument you're making).

I couldn't find any discussions of this in a style guide (it's difficult to locate non-prohibitions) and I haven't found a corpus that allows searching on open/close quotation marks, so I don't have any hard evidence to support my answer. I can't think of a specific example at the moment but I'm certain I've seen the use of closing then opening quotation marks with no intervening punctuation, though it is unusual—perhaps it's avoided as strange looking or ambiguous, but more likely because an author needs to add intervening words or punctuation to make the sentence work correctly.

All that being said, I think it would also be acceptable to use a second ellipsis, as in

"We the People of the United States...do ordain and establish this Constitution for the United States of America...in Order to form a more perfect Union"

because this too is true to the original and each of the phrases between ellipses is a direct quotation. This option versus the closing then opening quotation marks is probably more an editorial decision, but it's difficult to imagine anyone claiming that either was an improper citation of the original.