The two types of cross-prossessing are completely different animals. C41 processed as E6 is the rarer of the two, and in my opinion is potentially a very beautiful technique (I've noticed a slight resurgence of its popularity in Europe this summer). You can a color positive image, but with the orange masking layer still intact. It's inherently low contrast, with an orange cast, basically. It never really caught on because of the orange. I've experimented with it very little, but I always processed normally, and I'd expect pushing would be even better. You wouldn't want to pull, the look is very flat as it is.
"Normal" cross processing, making C41 negs of E6 emulsions, very definitely benefits from overexposure and pull, thereby flattening out its highly excessive contrast, thus allowing you to print for greater saturation of the slightly false color throughout the scale. If you do not overexpose and pull, then your highlights or shadows are guaranteed to be off the scale, unless the subject and lighting conditions you're shooting are VERY flat. Figure RA-4 paper's got about a 4 1/2-5 stop range, and a cross-processed neg is likely to have a range of 6-7 stops or so (though it doesn't keep a lot of detail throughout, the highlights tend to block up real fast). If you can pull it back a stop, you'll be close to the range of the paper you're printing on, as well as being closer to the density range of a scanner you might be scanning to.
The look of E6-to-C41 chemistry cross-processing is well-known. You get a high-contrast, grainy look of saturated, false color, where the color palette tends to restrict itself primary colors, basically. What would normally be rendered as subtle shades of cyan to blue become almost pure blue or pure cyan, for example. And then in the printing stage you can dial in a color cast on top of it, if you so desire. I recommend that you begin with E100S or E100SW as your first emulsion to experiment with, and then try some others if you want. I and many others I know had the most success with these two.