This is a problem with domain switching.
I was not able to reproduce with the URL you provided which used the www address for your site, I wont make your URL public, but for example sake:
using that address the Items went into the cart without a problem. But, using the non www address:
that is when i was able to reproduce the problem when it redirected to the cart page using the www address:
The cart items are stored in session variables. the browsing session is identified by domain. as far as sessions are concerned, the www and non-www address for your site are 2 separate sessions.
to prevent the problem, use htaccess to force all traffic to use the www address: