  In this paper a generalisation of the notion of polarity is exhibited which
allows to completely describe, in an incidence-geometric way, the linear
complexes of $h$-subspaces. A generalised polarity is defined to be a partial
map which maps $(h-1)$-subspaces to hyperplanes, satisfying suitable linearity
and reciprocity properties. Generalised polarities with the null property give
rise to a linear complexes and vice versa. Given that there exists for $h>1$ a
linear complex of $h$-subspaces which contains no star --this seems to be an
open problem over an arbitrary ground field --the combinatorial structure of a
partition of the line set of the projective space into non-geometric spreads of
its hyperplanes can be obtained. This line partition has an additional
linearity property which turns out to be characteristic.
