I'm reviewing some classical analysis using a text from the 90's. When I took a course using Rudin last year, I never learned this terminology.
If $x\le y$, then $x+z\le y+z$. (compatibility of $\le$ and $+$)
If $0\le x$ and $0\le y$, then $0\le xy$. (compatibility of $\le$ and $\cdot$)
It appears, based on the first definition, "compatibility of $x$ and $y$" means performing the same amount of the $y$-operation on both sides of the $x$-relation preserves the relation. If this is the correct definition, why isn't the second definition phrased as "If $x\le y$ and $z\ge 0$..."? Also, do we use "compatibility of $x$ and $y$" and "compatibility of $y$ and $x$" interchangeable, or is order of importance?
The comaptibility is a concept which relates relations and operations both defined over the same set. In general the following definition of compatibility is assumed: given any set $X$, a relation $\sim$ over $X$ and an operation $\ast$ over $X$, we say $\sim$ is compatible with $\ast$ iff $$\forall a,b,c,d \in X \big(a \sim b \land c \sim d \Rightarrow a \ast c \sim b \ast d\big)\,.$$ Now you have $X = \mathbb R$ (I think), $\le$ instead of $\sim$ and the multiplication $\cdot$ instead of $\ast$. If you choose $b$ and $d$ equal, you will obtain the first compatibility you mention. The second is simple: let $a$ and $c$ be equal to $0$.