Is there a name for relations with this property? Is there a name for relations $\rho : X \rightarrow Y$ such that for all $x,x' \in X$ and all $y,y' \in Y$ we have that the following conditions $$xy \in \rho$$ $$x'y \in \rho$$ $$xy' \in \rho$$ imply that $$x'y' \in \rho?$$
Here's a couple of alternative characterizations.


*

*Define $\rho(x) = \{y\in Y \,|\, xy \in \rho\}$ for all $x \in X$. Then for all relations $\rho : X \rightarrow Y,$ we have that $\rho$ has the property of interest iff for all $x,x' \in X$ it holds that either $\rho(x) = \rho(x')$ or $\rho(x) \cap \rho(x') = \emptyset$.

*Call a relation $\kappa : X \rightarrow Y$ Cartesian iff there exist $A \subseteq X$ and $B \subseteq Y$ such that $\kappa = A \times B$. Call two relations $\kappa$ and $\kappa$' strongly disjoint iff their images are disjoint, and their "left-images" are also disjoint. Then for all relations $\rho : X \rightarrow Y,$ we have that $\rho$ has the property of interest iff it can be expressed as a strongly disjoint union of Cartesian relations $X \rightarrow Y$.
A few observations:


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*If a relation has the property of interest, so too does its converse.

*Every function has the property (and thus, so too does its converse).

*If two relations have the property, their composition does, too; thus, we obtain a category.

*The property is preserved under arbitrary strongly disjoint unions.

 A: If the relation is seen as an undirected bipartite graph with edges joining $X$ and $Y$, the axiom is that a path of length 3 closes to a cycle of length 4.
In this form it is easy to check that "are both related to the same element in the other set (or equal)" is an equivalence relation on $X$ and on $Y$, and that if $x$ is related to $y$, all elements of $X$ equivalent to $x$ are related to all elements of $Y$ equivalent to $y$.  This is the condition for there to be quotients of the sets $X$ and $Y$ on which the relation descends to a one-to-one correspondence.
This can be expressed by saying that $\rho$ is/induces an "identification between quotients" of $X$ and $Y$.  That property is not closed under composition as far as I can see, but the other properties are true.
A: Such relations are called rectangular in Section 5.2, page 669 of Andrei A. Bulatov, Víctor Dalmau, Towards a dichotomy theorem for the counting constraint satisfaction problem, Information and Computation, Volume 205, Issue 5, May 2007, Pages 651-678. They in fact define the property for higher arities, but it simplifies to your definition for binary relations.
A: In regular categories, relations with this property are called difunctional or sometimes Mal'cev relations (or any other transliteration of the name). This terminology is used for example in the paper "Diagram chasing in Mal'cev categories" by Carboni, Lambek and Pedicchio, where it is also proved that a regular category is Mal'cev if and only if every relation is difunctional.
In the category of sets (or more generally any pretopos, see this paper), one can prove that a relation $R\subset X\times Y$ has this property if and only if it is a fibered product
$$\require{AMScd}\begin{CD}R @>{\pi_Y|_R}>> Y \\ @V{\pi_X|_R}VV @VV{g}V \\ X@>>{f}> Z.\end{CD}$$
In fact is is then the fibered product $X\times_{X\sqcup_RY} Y$, where $X\sqcup_RY$ is the pushout of the span $X\leftarrow R\rightarrow Y$, which can be obtained as the disjoint union of $X$ and $Y$ quotiented by the equivalence relation $R'$ defined for $x,x'\in X$ and $y,y'\in Y$ by : 


*

*$(x,x')\in R'\Leftrightarrow (x,x')\in R\circ R^°$ or $x=x'$

*$(x,y)\in R'\Leftrightarrow (x,y)\in R$

*$(y,x)\in R'\Leftrightarrow (x,y)\in R$

*$(y,y')\in R'\Leftrightarrow (y,y')\in R^°\circ R$ or $y=y'$.


Using this construction it looks like we can obtain your second point.
