Monotonicity on metric? (Is there a proper name for this characteristic?) Let $(X,d_X), (Y,d_Y)$ be metric spaces. Let $f:X \to Y$. Suppose $f$ satisfies that
$$
d_X(p) \leq d_X(p') \implies d_Y(f(p)) \leq d_Y(f(p'))
$$
where $p = (x_1,x_2), p'= (x'_1,x'_2), f(p) = (f(x_1),f(x_2))$. 


*

*I want to know the name of this property if there is any.

*Are there related works?

*What would be a non-trivial equivalent characteristics of $f$?


Example: $(\mathbb{R},d)\to (\mathbb{R},d): x \mapsto x^r$ with $r$ odd.

Such maps appear under various names: 


*

*ordinal embedding (see Local Ordinal Embedding or Ordinal embeddings of minimum relaxation: General properties, trees, and ultrametrics)

*monotone embedding (see Monotone maps, sphericity and bounded second eigenvalue )


Other names appear in applied literature, the aforementioned "Local Ordinal Embedding" paper by Terada and Luxburg has a list of   references going back to 1960s. 
The "Monotone maps" paper by Bilu and Linial is the most mathematical treatment of the subject to my knowledge, although it's classified as theoretical computer science.
I don't think there is any "nontrivial characterization" other than the definition you stated. What's a characterization of homeomorphisms? Being a homeomorphism, of course. Same for bi-Lipschitz maps, etc. The interesting problems are finding obstructions to such embeddings, and constructing embeddings when they exist. 
