Dark matter doesn't interact with electromagnetic radiation, but it, at least, participates in gravitational interactions as known from the discovery of dark matter. But does dark matter exist in a spiral shape around our galaxy?
In current cosmological models, the Milky Way resides in a 'halo' of dark matter. Halo is a technical term - in this case, it means a spherically symmetric collection of dark matter. Since dark matter is not self-interacting and does not interact with other matter, it doesn't experience any sort of collisions or friction, and therefore never flattens out into a disk the way normal (baryonic) matter does. So, dark matter does not trace out a disk and does not follow spiral arms.
The 'spiral arms' do not contain a lot more matter than the 'not spiral arms'. Rather we pick these out as they glow bright with new star formation. The density distribution is fairly constant throughtout the disc.
According to the latest modelling the dark matter component is a spherical shape on small scales but forms threads between conglomerations centered on galaxy clusters. Given dark matter forms 5x the mass of light matter it makes sense to reverse this and say light matter congregates within dark matter structures.
Disclaimer: I don't do dark matter, I don't work with anyone whose does and I haven't read enough papers to make an difference. I have seen a colloquium and a couple of seminars by people who do do dark matter.
Take what follows with that in mind.
Dark matter either not having been observed or having served up exactly one unconfirmed observation there is no widely agreed theory of its behavior yet.
There are models of cosmological evolution that posit certain traits for the dark matter and when run forward from the presumed conditions of the early universe give something that resembles the universe we see today.
In those models, the dark matter distribution is roughly spherical and rather larger than the luminous portions of the galaxy.
The main way I know of for studying dark matter distribution (I actually just wrote a grant application for research relating to it! :P) is gravitational lensing. Depending on the geometries involved, one can fairly precisely model dark matter distributions by looking at distortion of "nearby" (as far as arc-seconds on the sky, not actual physical distance in space) galaxies by weak gravitation lensing. A nice example (with a pretty picture) is available here.
Since dark matter is 96% of the matter in the universe, the question is complicated. In fact we're distributed around dark matter, if anything. That's a bit in general, and is covered a bit in analysis of Millennium I and II (which are incredible search for Cosmography of the Universe and stick with the twenty minutes video). Schneider More appropriately there has bee work to constrain the model of the exact subject of your question [Vera-Ciro].
The shortest version is no, the shape of dark matter halos is for the most part elliptical, with the eventual relaxation into a spherical halo as time runs.
Disclaimer: this is not my particular field, but I've been reading recent papers on it because it provides quantitative data for something I'm thinking about.
The best estimate of the shape of a dark matter halo from the dynamics of the Milky Way is that its dark matter halo is rugby ball shaped with the long axis of the rugby ball like a stake running through Sag A* perpendicular to the plane of rotation.
In general, efforts to describe the shape of dark matter halos favor what is called an "isothermal" profile as opposed to the NFW profile which is what you would naively expect doing essentially a statistical mechanics type analysis.