Brief answer: No.

The header/footer page determination of odd/even is the page number assigned to the page in the page number formatting.

The headers/footers in one Section can be linked to those in the previous Section. If linked they will be the same but even being the same, because of fields can display very different things. The most common fields to make these changes in what text is shown even though the underlying header/footer is the same are the StyleRef Field and the Page field. Another very useful field in a multi-section document is the SectionPages field. Field codes in Word - Word - Office.com

Header/Footer Settings Recap

However, you can set Sections to start on ever or odd pages which may get you what you want.

Typically, a new logical section in a document (like a chapter) will start on an odd-numbered page. There may be a blank page before it. This way the beginning of the chapter starts on the left hand side of a document printed on both sides. The start of a Word Section can correspond to the beginning of the logical section.

Another reference that may help is: How to set up a document with front matter 2007