Is matter - dark matter symmetry possible? The gravitational lensing evidence suggests massive and relatively dense accumulations of dark matter in space. However, dark matter collapsing to dense regions is unlikely due to energy conservation. Could there be a different explanation of how dark matter behaves? For example, is a matter - dark matter symmetry conceptually possible? 
Consider the following scenario. For each type of a matter particle, there exists a corresponding dark matter particle. And also, for each type of a normal interaction, there exist a corresponding dark matter interaction. This way the dark matter behaves the same way as regular matter, perhaps with dark stars forming and emitting dark photons. However, the only interaction between matter and dark matter is gravitational through the curvature of common spacetime.
This way dark matter stars could co-exist with normal stars in the galaxy explaining the galaxy rotation. Also, some galaxies could be made predominantly of dark matter explaining the gravitational lensing.
Is there any reason why, for example, two diffeent electromagnetic interactions could not exist for two different types of matter without interacting with each other?
 A: The problem with your hypothesis is that observation strongly suggest that dark matter behaves as a pressureless fluid: this is compatible with your hypothesis only if the interactions between dark matter particles are very much weaker than that between their standard partners. For long, this was only a hypothesis which was indirectly well confirmed by the success so-called $\Lambda$CDM cosmological model, which assumes a cold dark matter and a cosmological constant. So in addition to being weakly interacting, dark matter particles are slow in this model. By success, I refer to the inhomogeneities of the CMBR and of galaxy cluster distribution. But recently, [SR11] claim to have measured the equation of state for dark matter by combining weak gravitational lensing and rotation curve for clusters of galaxies. Their best fit is compatible with a pressureless dark matter.
So, of course, that does not rule out your hypothesis, but at least it shows that the symmetry between the standard particles and dark matter particles is extremely broken. At the very least, your dark matter world would be nothing like the standard one: with such weakly interacting particles, no way to have stars or any such assembly. So far the dark matter world is totally compatible with being a dust.
[SR11] Ana Laura Serra and Mariano Javier L. Dom ́ınguez Romero. Measuring the dark matter equation of state. Monthly Notices of the Royal Astronomical Society: Letters, 415(1):L74–L77, 2011. [Free access at arxiv]
