I have answers.
By the way, that sexy photo you see everywhere of the USS Birmingham doing a full blow and breaching like a whale was shot on sea trials and my friend Mark is driving the boat. He said they only performed that maneuver twice.
The helmsman steers the boat. His yoke has control of the rudder and stern planes.
He is the one actually turning the boat via the rudder. And controlling the ascent or descent of the boat via the stern planes.
Another man controls the fairwaters or bow planes. On the Birmingham these are on the sail.
They adjust the pitch of the boat known as the "bubble". The fairwaters act in conjunction with the stern planes to ascend or descend.
However, the boat can dynamically dive with the nose pointed up or ascend with the nose pointed down but not nearly as efficiently.
A third guy trims the boat by adjusting the ballast tanks. He's pretty busy all the time.
A submarine can dive or surface statically, i.e. with no forward motion or force on the planes.
The ballast tanks accomplish this. He can also aid in ascent or descent by flooding or blowing the tanks fore and aft in a dynamic dive.
There is also the capability to roll the sub and according to Mark, this may be done only under extreme circumstances.
In emergencies or in severe evasive maneuvers. It is not a normal component of steering the boat and most of the time the ballast guy
is making adjustments to correct the roll that steering maneuvers can induce.
He said that the left and right dive planes do not work in opposition. In fact they are pinned together.
So the sub cannot make an "aileron roll" like an aircraft. He added that you can go to the back of the boat,
physically un-pin them and hydraulically operate them independently but that is probably for maintenance reasons.
So the answer to the OP's question is that the sub is not normally banked to turn.
The roll is a result of the turn as performed by the rudder and is dependent on the speed and tightness of the turn.
The roll is induced by the uneven action of the water against the planes in the turn.
"Cambridge Ocean Technology Series 2 Concepts in Submarine Designs", page 176.
"When the submerged submarine turns in a horizontal plane, the action of the bridge fin is more complex. Efforts to reduce its drag results in a streamline shape, rather like the airfoil section of a wing of low aspect ratio, which will generate lift when at an angle of attack to the flow. Those circumstances arise in a turn when the hull is in general drift motion and because the sideways lift force on the bridge fin acts well above the axis of the submarine a heeling moment results. The drift angle on a submarine in a tight turn can build up quite quickly and the dynamics of the corresponding heeling moment from the bridge fin can cause a 'snap roll' which although it is inward on the turn can be far larger than is appropriate for a banked turn effect."
It's actually more of an undesirable phenomenon.
Page 157.
Roll Aspects
"It is not unusual to specify any requirements for control of the roll motion, though the designer needs to know the caused of the phenomenon termed 'snap roll', which we describe later, and what steps can be taken to minimize it. It is the practice to lay down a limit on maximum roll angle - 50 degrees is not untypical - to which equipment and machinery on board the submarine have to be designed. Whilst this is set as a design limit it is not acceptable for normal operation."