How to remove headers and footers for pages between chapters?

Use the emptypage package.

You could redefine \cleardoublepage:

\makeatletter
\renewcommand*{\cleardoublepage}{\clearpage\if@twoside \ifodd\c@page\else
\hbox{}%
\thispagestyle{empty}%
\newpage%
\if@twocolumn\hbox{}\newpage\fi\fi\fi}
\makeatother

The fancyhdr package deals with this topic in its documentation. Have a look at section 15 Those blank pages on page 15.

I believe this is the default behaviour for the memoir class.

It can be further (bizarrely) tailored by, for example

\cleartooddpage[\thispagestyle{companion}]

or

\cleartoevenpage[\thispagestyle{Ruled}]

but, sadly, not for \cleartorecto or cleartoverso

---

Slightly more diligent research show this: \cleartorecto and cleartoverso include this call:

 \thispagestyle{cleared}

The cleared memoir page style is defined by:

\aliaspagestyle{cleared}{empty}

and all is nicely described in section 7.2, "Page Styles" of the memoir manual (memman.pdf)