I have done the experiment Anti-Lorentzian Motor (pdf), (word) that showed a magnetic force parallel to the current turning a rectangular coil. Someone contends that this movement can be explained with Lorentz force. I agree that there is confusion. He also says that in order to avoid Lorentz force the coil should be circular. I find this idea very good and put a circular coil in a magnetic field to see if it rotates. And it rotates.
Please read the article at
Circular motor driven by tangential magnetic force http://pengkuanem.blogspot.com/2014/02/circular-motor-driven-by-tangential.html
or document word with the video integrated