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I run in to the same (and other) issue(s) after updating to Joomla 3.3.6.

First issue was that the relative URL just showed a blank page so "www.mysite.com", this seems to be a known Joomla problem.
Just add "/index.php" to the URL and it will work again.

Second issue was that i wasn't able to save any changes in the NuModusVersus template.
I used the quickstart package for Joomla 3.1.

1. Setup the site with the quickstart package
2. Update to the latest Joomla version (3.6.6)
3. Update component/plugins etc.
4. Purge cache
5. Check if your NuModusVersus template is working correctly, if not...
6. Change the default template to an other template
7. Remove/Uninstall the NuModusVersus template
8. Reinstall just the template package (1.0.6)
9. Change the default back to the NuModusVersus template
10. Purge cache

This fixed my issues :-)

Thank you for sharing Chris.

Actually the step n.7 was not necessary you can install the latest version on top so you won't lose any settings.
Other than that everything else looks perfect.