Everyone has pretty much stated the facts, but there are a few things to remember.
Pushing E6 is accomplished by varying the first developer (we call it the MQ). Anyhow, up to a point, what happens is that you increase fog in the MQ, and as fog goes up, dmax and apparent camera speed (EI) goes up and you are pushing the film while lowering the dye dmax by raising the negative silver fog. A pull lowers negative image fog and raises Dmax giving slower EI due to more dye forming. Basically the E6 MQ is a very foggy developer and the emulsions are very foggy as well. The reason is given above.
This can either raise or lower contrast of the dye image depending on film.
Basically then, by varying the MQ time, you push the film first or pull it with little color change but a large change in dmax and a moderate change in EI. Beyond that point, color shifts begin to take place on the axis defined by the film that you are working with. It varies with film and the activity of the 3 color layers. To fix the problem, you have to experiment with the color developer. This is done either by changing time in the color developer, or by changing pH, or both.
Hope this helps. Have fun.
Oh, I said just about each E6 film reacts differently, well, each E6 kit reacts somewhat differently as well. Some of the kits don't use HQ-monosulfonate in the MQ and therefore the first developer behaves differently than the genuine article. Some don't use the right solvents and halide balances as well, causing other problems. So, be carefule, all E6 kits are not born equal.