( Bold part of answer is the real answer; the rest is just background )
Rabbi Ribiat writes in The 39 Melochos (in the corrected edition, Misrad Hasefer 2004) [Section 2, Chapter II, section G; page 214]:
If a Melocho is performed in what is Halachically recognized as an
irregular mode, it may be considered a "Shinui" (שינוי). ....a Melocho
performed in an awkward or backhanded manner does not conform with the
character of the Melochos in the Mishkan.
In the footnote there (number 146), he writes that
כל שאין דרך לעשותן הכי או שקשה לו לאמן ידו לעשות המלאכה נכונה הוי שינוי
Anything that is not usually done in this fashion, or if it is difficult to train one's hand to do it properly, that is a
In that footnote, he cites the Chiddushei HaRan to Shabbos 103a (sv אמר אביי, at the end)1, as proof to this idea. When someone writes on Shabbos, he is only liable on a Biblical level when he writes with whichever hand he usually writes with; but someone who carries from one domain to another is liable whether he used his right hand or his left -- it's normal to carry with either hand, and doesn't take much training or getting used to in order to do it in an efficient manner. Writing, on the other hand, is unusual to find righties writing with their left hands, and it's pretty difficult to teach yourself how to be ambidextrous (believe me, I've tried ;).
1 Note: You won't find this in the regular blue, Mossad Harav Kook Ran on Shabbos. Many thanks to Fred for providing the link in comments. :)