Which spelling is correct: "Re-order" or "Reorder"

Reorder.

With re- words, you should use ‘re-’ (with a hyphen) if the next word begins with an ‘e’ or a ‘u’ (when not pronounced like ‘you’). Otherwise, don’t hyphenate. It’s therefore re-examine, re-urge, re-entry and re-elect, and reuse, reunion, reorder, reinforce and redevelop.

Source: http://www.proofreadinglondon.com/blog/to-hyphenate-or-not-to-hyphenate

In general, try to avoid putting hyphens into words formed of one word and a short prefix

Source: http://www.economist.com/style-guide/hyphens