Nice question Matthew,
As far as we know the answer is no. Astronomers have looked at the spin and rotation directions of a large number of galaxies but the net angular momentum is zero, within statistical uncertainties. You can always find a few local anomalies but, again, consistent with statistical fluctuations.
(published on 01/12/2012)
(published on 10/11/2010)
I guess that you're referring to the effect described in this article:
http://physicsworld.com/cws/article/news/2011/jul/25/was-the-universe-born-spinning . The paper it describes said that, looking northward from Earth, a slight but statistically significant majority of the galaxies are seen to rotate counter-clockwise. However, another survey of a large number of galaxies found no effect: http://adsabs.harvard.edu/abs/2008MNRAS.388.1686L. I can't find any paper confirming the surprising results of that first one. So it's hard for outsiders (or even an astrophysicist who helped out here) to be sure of whether the effect is real.
The standard Big Bang pictures don't include any overall rotation, although obviously local accidents make galaxies, stars, etc. revolve. If the non-random looking rotation effect turns out to be real it might involve just some corrections to a messy picture of how large-scale swirls and counter-swirls got going as the early universe evolved. There's some discussion of this issue in a paper posted on the Arxiv (not refereed): http://arxiv.org/pdf/1201.5180v3.pdf, but I'm not familiar enough with the area to evaluate it. That paper describes some rotation effects that extend farther than expected but not throughout the universe.
There's always a slight chance that some surprising result might overturn central parts of our understanding, but usually strange results turn out to be somehow off.
(published on 10/19/2013)