Is dark matter influenced by gravity of baryonic matter? As far as I know galaxies and clusters form along so-called "galactic filaments" which are influenced by a dark matter web that organises them in to this position in the universe. So it looks like dark matter attracts baryonic matter by a kind of gravity. But is it also true the other way around, and that 'our' mass has gravitational forces on dark matter?
And if not why does it only go one way?

In a study scientists have observed the four colliding galaxies of cluster Abell 3827 and discovered that one of the envelopes of dark matter in the galaxy behind it encloses. The backlog in the system stands at 5,000 light years (50,000 million million kilometers - the NASA space probe Voyager would do 90 million years to cover that distance.
Expected to create some lag between a galaxy and its associated dark matter when dark matter affects itself, however small, by other forces than the gravity. 

See also https://www.youtube.com/watch?v=qtfTUPhXutQ and
http://spacetelescope.org/news/heic1506/
Dark matter attracts baryons not only by a "kind of gravity", but simply by gravity. Everything that has mass (even if it's only relativistic mass like light) attracts everything else by gravity, and thus baryons also attract dark matter.

The reason dark matter "organizes" galaxies is just that there is much more of it, roughly 7 times more, to be specific. So, on large scales, the dynamics of galaxies are dominated by dark matter. But because baryons can dissipate energy, it may collapse in clumps — such as molecular clouds — which are hence dominated by baryons.