The rudder of course always controls yaw, but our control of yaw has 2 separate jobs -- of controlling heading, and controlling sideslip. If the thrust is symmetrical, they are tantamount to the same thing, and we don't really think about it.
But with asymmetrical thrust, it starts to matter. Let's say we lose the right engine. If we do nothing about it, it wants to yaw right due to the thrust from the left side. So we naturally stop the right yaw with left rudder. The clockwise yaw torque from the thrust, is balanced by the counterclockwise yaw torque from the rudder. Therefore heading is steady.
But the rudder, in addition to producing a counterclockwise torque, is also producing a force to the right - with nothing to counter it. Therefore, we will drift to the right. Heading and direction of flight no longer match, which is a slip. That is why zero heading change no longer equals zero slip, as it does in normal flight. To fix this, we counter that rightward force with a leftward force: a horizontal component of the lift vector, obtained by banking slightly left. Now the left and right forces match, and nothing is pulling us sideways. Ergo, heading and flight patch match, and slip is zero. (But the ball is to the left).
But here's a crucial side effect - and the answer to John T's question. There's another contributor to the yaw torque on the plane, besides the rudder and the asymmetrical thrust. It's any sideways air hitting the back of the plane, i.e. the slip itself. It too must be countered with rudder to maintain a heading. Recall that in symmetrical-powered flight, it takes a constant rudder input to maintain a slip. Without it, the sideways-hitting air will turn the tail into the "new" direction and the slip will disappear.
Well, in our right-engine-out scenario, if we go back to the no-correction situation, the air will come from the right and yaw (and boat-turn) the plane to the right. So we need left rudder to prevent that. If we eliminate the slip by banking slightly left, this will disappear! If we bank more than the required amount to the left, we'll then slip the other way and will need right rudder. <-- But, the rudder inputs laid out in this paragraph are only for the slip; they're in addition to the rudder input for the asymmetric thrust. The actual rudder input is the sum of both. In case you're starting to get lost in the weeds (I am, in writing this) let's try a table format, with sample bank angles and made up "units" of rudder travel.
Code:
[br]Condition ruder for rudder for total[/br]
[br] ass. thrust slip rudder[/br]
[br][/br]
[br]No bank 5 5 10[/br]
[br]2 deg. of bank 5 3 8[/br]
[br]4 deg. of bank 5 1 6[/br]
[br]6 deg. of bank 5 -1 4[/br]
I had to do weird things, and it still won't show up quite right.
Bottom line is, that the more you bank to the left, the less left rudder you need. You can get so much effective left yaw torque by banking, (at the cost of increasing drag) that you can artificially lower the Vmc. This is why the certification limit for Vmc is a maximum of 5 degrees of bank. Any more yaw authority must be obtained by rudder.