What is the web site url? My guess is that they are coming to the page with http://www.yourwebsite.com and being redirected to https://yourwebsite.com or vice versa. Since it is technically a different session depending on whether you have the www or not, the cart is cleared and they get redirected.
It has nothing to do with the pages themselves... the solution is to use your .htaccess file to either force it to always or never use www.