Is Nm the same unit of torque as mN? A couple of days ago, I noticed that the torque unit used by my teachers is $mN$, and while reading on the internet it came to my notice that in all textbooks the official unit is $Nm$.
I asked one teacher about it and he insisted that I'm wrong, and while I told him that I read it on Wikipedia, he said that the sources or references used by Wikipedia aren't necessarily correct and I think I agree with him on that.
I checked my book and the only time it's mentioned is while discussing string torsion (I explained how it appears at the end of the post) but while solving problems, all our physics teachers us $mN$.
All of my teachers convince us using the idea that torque should be distinguished from energy since their units have the same dimensions and they represent different quantities (and I agree on this one) and that torque is the cross product between position vector and force to further support their point that it must be $mN$ and not $Nm$ even though the second point doesn't make any sense.
I kept looking online and eventually found the SI units official brochure published by the International Bureau of Weights and Measures and it clearly states that the unit must be $Nm$ and it couldn't be wrong (since it's the official units reference by definition).
My problem is that my school textbooks, while not in English, write its equations in English letters and notation (also left to right) so it couldn't be a matter of translation.
I'm not even trying to argue with him (because that's impossible, even if I had proof, he's too stubborn)
and in fact, he told me to look into more trustable references (he suggest old French textbooks/literature since he thought they were more dependable on than others even though I couldn't find any and they're probably outdated by now).
So is this choice of units purely conventional, or does matter mathematically, or are both units correct?
 A: The system of units has nothing to do with the physics behind the formulas, so it's pure convention. The SI is just a special system of units, which was made to be a standard, and maybe to make calculations easier (as there is simple transition rules between the different scales like m and km). (Also the SI was designed for the everyday life, I think, as the base units like m, s, etc. are well fit in in everyday acts like walking and talking. I.e. one's step length is around 1 m, and you can say a few words in 1 s.)
Technically mN (meter-newton) is not wrong, but a bit confusing for most physicists, as they will read that millinewton. Probably (one of) the most common notation(s) for torque is Nm, but foot pound is also commonly used, as mentioned in the comments.
In general there is lots of different notations for the same quantities, depending on the context. (1 angstrom = $10^{-10}$ m  and 1 eV (electronvolts) $= 1.6\cdot10^{-19}$ J is used in atomic and molecular physics for example.) In case of torque for example, if somebody use a different unit for force (instead of N), than the units of torque will follow that convention, as it's an inherited/derived(?) quantity.
I think the Wiki page is mostly reliable about the (SI) system of units. (In general the English Wikipedia is reliable for studying, but obviously not for research.)
Edit: (One question about the radians was removed from the original post, but I leave the answer here. I hope it's not a problem.)
And about the radians... In SI it is considered as 'dimensionless' (as in SI base units its dimension is [m/m]=[1]), but in many cases it is straightforward to use rad to indicate that there is an angle in the equation. And also mrad (milliradian) is often used.
A: The unit for torque is $force \times distance$. So Newton-meters (N-m) is what it would be. However, since it is multiplication so meter-Newtons isn't really wrong, but no one says that.
So your teachers say Wiki isn't accurate. Okay. What did they say about every textbook out there? I'm curious if any of them have written at textbook and what they used. I find it amusing that your professor completely changed the subject by saying French textbooks were more reliable, thus weakly implying they would use mN, but did not actually say they did (and definitely showed no examples off his shelf). He basically changed the subject and dodged your entire question when he did that.
A: There is a formula that says that the work done by a torque is torque times angle. The best units for torque are work/angle (Nt-m/radian).
A: When does one care about the units of torque? When one buys a torque wrench.  A quick market review aka first page of Google results finds the following units in product catalogs (American market, obviously): ft.-lb., ft.-lbs., in./lb., ft./lb., ft./lbs., Nm.
No mention of mN.  On the other hand a number of those units are wrong if symbols are taken to have their usual meanings.
So while your teacher certainly has reasons (cross product order as others pointed out) to chose the (ugly, misleading to anyone not using the same textbook) mN over Nm, I believe they would be doing you a better service by preparing you for a world where things don't follow a unique prescription and where even basic mathematical concepts are messed up all the time instead of insisting on a rule that doesn't exist outside their textbook.
A: Just like $2\times3=3\times2$, There is no difference between newton-meters and meter-newtons. They're two different ways of saying the same thing.
Probably your book is trying to avoid confusion when you learn about energy, which is also measured in newton-meters, although we normally rename the unit, when referring to energy, as joules.
You should go ahead and call the unit of torque "meter newtons" in your class, because that's what your instructor expects. But be prepared to see other people call it "newton meters". I'd even strongly recommend using newton-meters anywhere except in your class, since the unit ${\rm m\cdot N}$ (meter-newtons) is much too easily confused with $\rm mN$ (millinewtons).
A: Looking at your username, are you by any chance in an area where the local language is written right-to-left? If so, then this could explain why your professors are writing mN: they (or whoever 'started this trend') are applying the RTL to the units in the equations as well.
Strictly speaking this is wrong, as others have elaborated, but it could be the local convention to do it like this. And it might even have arisen by accident, someone just happened to start doing it that way and in their mind the N was still coming before m, because of course you read text right to left. And then this person taught it to their students who then became the next generation of professors and propagated it further.
