When a website is included in the Internet Explorer Compatibility View List under the <Flash></Flash> section, which means it is to have Flash enabled on Windows 8 when rendered in the Modern UI, IE10 sends the wrong browser mode.
Specifically, IE10 uses the IE10 Compat View BrowserMode and therefore sends MSIE 7.0 in the User-Agent request header.
IE10 also logs something similar to the following in the develop tools console:
HTML1200: example.com is on the Internet Explorer Compatibility View List ('d:\Users\UserName\AppData\Local\Microsoft\Internet Explorer\IECompatData\iecompatdata.xml').
This happens for any site in the Flash section of the compatibility view list, which is not included anywhere else in the compatibility view list. It only happens on IE10 on Windows 7. The problem does not occur with IE10 on Windows 8.
For reference, Version: 10.0.9200.16618, Update Versions: 10.0.6