Yes, you should be able to see all your tasks on all your devices.  The most common situation that may be causing syncing issues is mismatched versions of Get It Done app.  Please make sure you are accessing the latest apps (version 5) on each device:

Just click the links on the device you want to use and it should take you to the app store or launch the web app.  

It might be worth deleting the old apps from your device to get into a clean slate.

If that doesn't work, try logging in and out of the app.  The logout button is found in the Settings of the app.

If you are still having issues, please write us a ticket!