  We introduce topological parallelisms of oriented lines (briefly called
oriented parallelisms). Every topological parallelism (of lines) on PG(3,R)
gives rise to a parallelism of oriented lines, but we show that even the most
homogeneous parallelisms of oriented lines other than the Clifford parallelism
do not necessarily arise in this way. In fact we determine all parallelisms of
both types that admit a reducible SO(3)-action. Only the Clifford parallelism
admits a larger group Surprisingly, it turns out that there are far more
oriented parallelisms of this kind than ordinary parallelisms. In particular,
we show that the SO(3)-invariant ordinary parallelisms constructed by Betten
and Riesinger are the only ones compatible with this group action. We also
study the rotational Betten spreads used in this construction and their
automorphisms. The automorphism group of the (oriented) parallelisms is always
SO(3), no matter how large the automorphism group of the non-regular spread is.
There are far more isomorphism types of parallelisms than types of spreads.
