\documentclass{memoir}
\usepackage{facingpages}

\begin{document}
Ok, we start with a couple of pages of non-interleaved text

We're on sheet \thesheetsequence
\newpage 

A second page of non-interleaved text. There should be a blank page
after this one.

\BeginFacingPagesLeft

Ok, now we're starting some interleaved text. 
This is the first page of the verso part.

Page \thepage, Sheet \thesheetsequence

\newpage
This is the second page of the verso part.

Page \thepage, Sheet \thesheetsequence

\BeginFacingPagesRight

This is the first page of the recto part.

Page \thepage, Sheet \thesheetsequence

\newpage %Page \thepage, Sheet \thesheetsequence\newpage

This is the second page of the recto part.

Page \thepage, Sheet \thesheetsequence

\BeginFacingPagesLeft

Now we're starting a second segment of interleaved text. 
This is the first page of the verso part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the second page of the verso part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the third page of the verso part.

Page \thepage, Sheet \thesheetsequence

\BeginFacingPagesRight

This is the first page of the recto part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the second page of the recto part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the third page of the recto part.

Page \thepage, Sheet \thesheetsequence

\EndFacingPages

Now we're beginning a non-interleaved segment. This is the first page.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the second page. There shouldn't be a blank page after this one.

Page \thepage, Sheet \thesheetsequence

\BeginFacingPagesLeft

Now we're starting a third segment of interleaved text, this time
with uneven numbers of pages. This is the first page of the verso part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the second and last page of the verso part. There should be two
extra blank page after this one.

Page \thepage, Sheet \thesheetsequence

\BeginFacingPagesRight

This is the first page of the recto part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the second page of the recto part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the third page of the recto part.

Page \thepage, Sheet \thesheetsequence
\newpage

This is the fourth page of the recto part.

Page \thepage, Sheet \thesheetsequence

\BeginFacingPagesLeft

Now we're starting a fourth segment of interleaved text, this time
with uneven numbers of pages. This is the first page of the verso part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the second page of the verso part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the third page of the verso part.

Page \thepage, Sheet \thesheetsequence


\newpage

This is the fourth and last page of the verso part.

Page \thepage, Sheet \thesheetsequence

\BeginFacingPagesRight

This is the first page of the recto part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the second page of the recto part.

Page \thepage, Sheet \thesheetsequence

\newpage

This is the third and last page of the recto part. Tere should be an
empty page after this one.

Page \thepage, Sheet \thesheetsequence

\EndFacingPages

Ok, one last page, non interleaved

\end{document}