WP Super Cache conflict with WPtouch

Hi. I have WP 3.5.1, WPtouch 1.9.6.1 and WP Super Cache 1.3.1 plugins, Atahualpa 3.7.12 theme. The problem is:
– Desktop browser sometimes shows a mobile version of the site (if that page was previously visited from a mobile device);
– Mobile device sometimes shows a mangled (without CSS) desktop version of the site (if that page was previously visited from a desktop browser).

The problem occurs with any of WP Super Cache’s three caching methods: “Use mod_rewrite to serve cache files. (Recommended),” “Use PHP to serve cache files,” or “Legacy page caching.”

To reproduce the problem:

Test #1:
(1.1) WP Super Cache Settings -> Easy -> Delete Cache; log off.
(1.2) Visit a post from Windows desktop Firefox 17.0.1 (displays desktop version as expected);
(1.3) Then visit same post from a mobile device (displays a mangled, without CSS, desktop version).

Test #2:
(2.1) WP Super Cache Settings -> Easy -> Delete Cache; log off.
(2.2) Visit a post from a mobile device (displays WPTouch mobile version as expected);
(2.3) Then visit same post from Windows desktop Firefox 17.0.1 (displays WPTouch mobile version!).

Any advice? I “think” I set up WP Super Cache as recommended for it to work together with WPtouch, but I may be missing something…

My WP Super Cache settings are:

WP Super Cache Settings -> Advanced: I checked the following
– Cache hits to this website for quick access.
– Use mod_rewrite to serve cache files //I also tested the other two caching methods
– Compress pages so they’re served more quickly to visitors
– Don’t cache pages for known users.
– Cache rebuild. Serve a supercache file to anonymous users while a new file is being generated
– Mobile device support.
– Extra homepage checks.

WP Super Cache Settings -> Plugins:
– WPTouch “Enabled”

I also made sure to do WP Super Cache Settings -> Advanced -> Update Mod_Rewrite rules.

http://wordpress.org/extend/plugins/wp-super-cache/