It's not not possible to move pages in a space or reorder pages in Confluence OnDemand.

There's no error that's reproduced on the screen or the logs, and the page will look like it got moved, but as soon you refresh it stays in it's original location.

Workaround:

Turn off the default space logo in Confluence Admin -> Default Space Logo (the URL should be like this (change <example> with the subdomain of your instance's URL) https://<example>.atlassian.net/wiki/admin/configuredefaultspacelogo.action). This will allow spaces to be moved.