Can I reorganize fields in a form that already has submissions?

Can I reorganize the fields in a form that already has submissions, without messing up existing data? I have discovered that some fields near the end would be better at the start. I don't want to change them, just move them.

I am concerned that if I make such changes, the all-submissions spreadsheet will no longer have all the responses under their correct headings. 

Repositioning the fields of the form will not affect any existing submission(s). The arrangement of field questions on the submissions page of your form will follow the order of the fields on the form. However, the changes you made in the order of fields on your form will not affect the order of columns in the integrated spreadsheet. As long as you are just reorganizing fields and not deleting them that would be fine. Existing submission data will not be lost.