Very weird indeed (bordering on impossible

).
Do you have this tag in the head section of all templates?
<base href="[[++sitr_url]]" />
If there is no .htaccess file in the root, the Friendly URLs System setting has to be turned off.
If that's not is, it might be a plugin or snippet that is corrupting the cache, or possibly a permission or ownership problem with the cache directory.
First, check the MODX error log. If there's nothing informative there, start disabling plugins one at a time (most recently installed first) until it goes away. If it doesn't, try disabling snippets in the template by putting a space between their opening brackets. You'll see the snippet tag in the output, but the snippet won't execute.
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PLEASE, PLEASE specify the version of MODX you are using.
MODX info for everyone:
http://bobsguides.com/modx.html