Proper use 

The adjustable spanner#Proper use section of this article still seems to be self-contradictory.

I found one discussion  that has some physics-based arguments that I found pretty persuasive.
One commenter says that the preferred way to use a pipe wrench is the opposite direction from the preferred way to use an adjustable wrench,
which may explain why so many people disagree on the "proper" direction to turn it.

Norm Abram, in "Adjustable Wrench",
implies that (unlike a basin wrench) an adjustable wrench often works adequately even when turned the "wrong way", such as when doing the "flip trick".
This may also contribute to why so many people disagree on the "proper" direction to turn it.

What exactly is the advantage of turning it the "right" way versus the "wrong" way?
Too often I hear that turning it the "wrong" way is "more likely to round off the corners" without any more details,
but I prefer an explanation that would make it possible to work out from first principles which way is the "right" way.
(I wouldn't be surprised to discover that there are advantages to each direction).