It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it immediate consequences to the general theory. For instance, uniqueness of limit of a sequence does not necessarily hold, bifurcations in the meaning of Cauchy sequences etc.
There are however, two classes of quasi metric spaces where things look very much like the symmetric case. Some definition: Let $(X,d)$ be a quasi metric space. Say that $X$ has vanishing asymmetry at $x\in X$ if for all $\epsilon >0$ there exists $ \delta >0$ such that $d(x,y)\le \delta$ implies $d(y,x)\le \epsilon$. Say that $X$ is of vanishing asymmetry if it has vanishing asymmetry at each $x\in X$. Say that $X$ is of uniformly vanishing asymmetry if for all $\epsilon > 0$ there exists $\delta >0$ such that for all $x,y\in X$ if $d(x,y)\le \delta $ then $d(y,x)\le \epsilon $.
Another way to view these conditions is as follows. Let $QMet$ be the category of all quasi metric spaces (with non-expanding maps). There is then a functor $QMet\to QUnif$, to quasi uniform spaces, sending a quasi metric space $X$ to the quasi uniformity generated by the entourages $\{(x,y)\in X\times X)\mid d(x,y)\le \epsilon\}$ (where $\epsilon >0$ ranges). Similarly, there is another functor $QMet\to QUnif$ sending $X$ to the quasi uniformity generated by the entourages $\{(x,y)\in X\times X)\mid d(y,x)\le \epsilon\}$. The equalizer of these two functors is the full subcategory of $QMet$ spanned by those spaces satisfying uniformly vanishing asymmetry.
Similarly, there are two functors $QMet\to Top$ and the equalizer of these is the full subcategory of $QMet$ spanned by the spaces of vanishing asymmetry.
My question is, are these classes of quasi metric spaces known and if so, what is the common terminology. Any references are welcome.
These spaces are the same as metric spaces; the standard metric you get by averaging $d(x,y)$ and $d(y,x)$ happens to be comparable to the quasimetric in these cases by an $\epsilon-\delta$ argument:
given a ball in the standard metric, it contains a ball in the quasi metric of half the radius.
given a ball in the quasi metric of radius $\epsilon$, choose a ball in the standard metric of radius $\min(\delta,\epsilon)$, and it is contained in the original ball.