Although the documentation for
TwoColumnRight clearly says, "Display the pages in two columns, with odd numbered pages on the right," Acrobat seems to treat this as meaning display the first page on the right. Here's an example with three even numbered pages. It's displayed right, left, right.
My best guess is that
TwoPageRight (which didn't work for me regardless) and
TwoColumnRight (which did) do not consider logical page numbers. So if you start with an even numbered page, you should use
(For what it's worth, although later versions of the PDF specification claim
TwoPageLeft appear in PDF version 1.5, the v1.5 spec doesn't list them. I tried changing the
\pdfminorversion but that had no apparent effect.)