Suppose that I have a continuous surjection $f: U \rightarrow V$ between two open subsets of the plane. Suppose that $f$ appears to be quasiconformal in the sense that there is a uniform constant $K \geq 1$ such that for each $r > 0$ and $x \in U$ there exists an $s$ such that the image of an $r$-ball centered at $x$ in $U$ contains a $K^{-1}s$ ball and is contained inside of a $Ks$ ball centered at $f(x)$. Is there anything at all that I can say about $f$? I've been struggling to find any literature on this topic as the quasiregular mapping theory that one might resort to when injectivity fails in the quasiconformality setting requires differentiability.


In the context I am studying it in (a rigidity problem in dynamical systems) I do have some other knowledge about this function $f$; it is Holder and it has certain homogeneity/scaling properties that force the quasiconformality type relation above. 



Non-injective continuous maps that appear quasiconformal
Such maps are called quasiregular. There is a highly developed theory of them.
Most of the classical theory you can find in the books of Yu. Reshetnyak, Space mappings with bounded distortion, AMS, 1989, and
S. Rickman, Quariserular mapings, Springer, 1993.


Probably the most fundamental fact about such maps is that they are open and discrete.


In dimension 2 their structure is quite simple: they are compositions of
complex analytic functions with homeomorphisms. Also if such a map is sufficiently smooth (in any dimension) then it is a quasiconformal local homeomorphism, and if smooth and defined in the whole $R^n, n\geq 3,$ then it is a global homeomorphism. But if you relax the smoothness conditions you can have many
interesting maps which are subject of study in this theory, the simplest example 
is $(r,\theta,z)\mapsto (r,2\theta,z)$ in cylindrical coordinates in dimension 3.


But a complete answer on your question "what can be said?" will have length of a large book. 


