pdflscape fails to rotate page containing only a float

pdflscape fails to rotate page containing only a float

I'm using pdflscape to place some landscape pages in a long portrait
document, because it contains some very wide floats.
However, when a float is so big that it needs its own page, it is no
longer rotated correctly:
The following code:
\documentclass{article}
\usepackage{pdflscape}
\usepackage{lipsum}
\begin{document}
\lipsum[1-4]
\begin{landscape}
\lipsum[5-6]
\begin{figure}
\lipsum[7-8]
\caption{Some text, rotated correctly.}
\end{figure}
\begin{figure}
\lipsum[9-12]
\caption{Some more text, wrong side up.}
\end{figure}
\end{landscape}
\end{document}
Produces that output:

How can I correctly rotate float-only pages?