When I open the file in Word 2010, one of the activated layout options is "Balance SBCS and DBSC characters" - unticking it fixes the problem.

I did a little searching and found that this option is for Asian character sets, which use DBSC, or double-byte character sets (DBCS - Wikipedia).

I think you've got Word set up to handle Asian characters, or perhaps you're actually in Asia, and your version of Word is set up to handle this type of thing out-of-the-box. In fact, looking at the XML again, I can see that Simplified Chinese is listed in settings and styles files.

When you've been creating brand new files using Word Online and you're not having the problem, maybe the online program isn't setting Asian-language attributes in the file, or maybe pasting into the only program is excluding the attributes that are causing issues on the desktop version.