The rotation is as you suggest, around the longitudinal axis, whenever the board is trimming or turning i.e. whenever it isn't heading at 90 degrees to the wave front. It doesn't happen with whole tunnels.
The tunnel provides lift at any angle of attack.. up down or sideways and anywhere in between in response to angle of attack in any direction. One of the main reasons why I use them is that they provide increased and more efficient planing lift in the tail without the use of a wide tail. There are in my opinion advantages in efficiency and in handling and control over a wide range of speeds using a narrower tail in combination with the tunnel.
With a wide tail the force required to get the board on the rail increases dramatically as speed increases, this changes the handling characteristics in response to changes in speed. The tunnel equipped tail behaves consistently at all speeds. The half pipe tunnel offers virtually zero resistance to rail to rail roll, as it simply rotates around the fore and aft axis of rotation of the board.
In addition the tunnel produces a fulcrum at the tail which vastly improves a board's ability to flex under load. Planing hulls have a high pressure area just behind the leading edge of the wetted surface area, but the pressure drops off quickly further aft and at the tail is is only a fraction of what it is further forward. So even a wide tailed board ( without horizontal fin area) has only one effective fulcrum, making it almost impossible to get flex under load from the rider. The tunnel's horizontal lifting area is below the bottom so it lifts in a higher pressure flow with a higher angle of attack, it also provides lift more efficiently than the hull because it has greater span efficiency. It can also be set up with a few degrees of positive lift against the hull, further increasing the force which it produces. So, with a tunnel on the tail the board responds flex wise to rider force in much the same way that a board does in a land flex test when it is suspended at nose and tail. Conventional fin setups don't do this effectively ( some conventional setups have a small amount of horizontal lifting area but this area is almost always quite far forward on the hull).
Tunnels also produce relatively high lift at very low angles of attack and at low speeds, so they give lots of drive as soon as the board catches the wave and also a precise lift and drive response to subtle direction changes.
Tunnels have the lowest induced drag of any fin type, this equals more lift and a better lift/drag ratio. They do have high wetted surface area, and since wetted surface area predominates at high speed whereas induced drag predominates at low speed, they are theoretically at a disadvantage at high speed. I haven't found that to be the case however as the lift which the tunnel provides increases by the square with greater speed, reducing load on the hull for lower hull drag and enabling hull wetted surface area to be reduced, thereby controlling to some extent the amount of wetted surface area used at any given speed.
The first time I tested a tunnel fin I was sold on them immediately. I put a four inch diameter tunnel on a favourite 91'" pintail which I'd been riding a single fin for a couple of years. The result was excellent, and serendipitously coincided with a rare 20 minutes of wedge tubes at a place called Shark Alley. handling was perfect, unlike all the flat planar 'hydrofoil' wings which I'd been testing.
Tunnels setups do have to be designed in harmony with the tail rocker though as the tail rocker affects the angle of lift... too much tail rocker and they can pull the tail down. Although the tunnel fin can be set at a higher angle of attack in the tail to overcome this only a small amount can be used as otherwise the exit area of the tunnel is too much greater than the entry area... this causes drag. Torturing the shape of the tunnel into an ellipse at the trailing edge can overcome this but results in 'toe out' on the tunnel sides, so it is easier to just get the rocker right.
Turbo tunnels are a different beast, I've tried 'tunnel on a stick' fins and they are basically rubbish. The axis of rotation of the tunnel is misaligned with the axis of rotation of the surfboard, this greatly increases rail to rail resistance. Also the turbo tunnel is set so that it pulls the tail down rather than up, this creates a lot of drag. a further problem with the 'turbo' is that the tunnel exit is much smaller than the tunnel entrance, again this creates drag, rather like towing a bucket with a hole in it.