  Veldkamp polygons are certain graphs $\Gamma=(V,E)$ such that for each $v\in
V$, $\Gamma_v$ is endowed with a symmetric anti-reflexive relation $\equiv_v$.
These relations are all trivial if and only if $\Gamma$ is a thick generalized
polygon. A Veldkamp polygon is called flat if no two vertices have the same set
of vertices that are opposite in a natural sense. We explore the connection
between Veldkamp quadrangles and polar spaces. Using this connection, we give
the complete classification of flat Veldkamp quadrangles in which some but not
all of the relations $\equiv_v$ are trivial.
