I can't explain why the type of section break changes as you insert
it, but it has been reported in other threads. A situation where Word
(or perhaps the printer driver) would "logically" change the type of
section break is when two different sections use different paper
formats; otherwise, the back and front of a page could have different
paper formats. Of course, in such a case it would force an odd page,
not an even page break.

Does Word allow you to change the "Section start" option on the Layout
tab of File>Page Setup -- after you've inserted the new section?

The behavior when deleting breaks can certainly be confusing. Part of
the difficulty is that a section break stores the formatting of the
preceding section; but it displays the type (Odd, even, continuous) of
the following section. For more about deleting section breaks, see:

Working with sections
http://word.mvps.org/faqs/formatting/WorkWithSections.htm