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It seems nicely symmetrical that opposite ailerons would be locked togther, so that when the left wing's ailerons move up, those on the right wing would move down, by the same amount and at the same speed.

Is this in fact generally the case on aeroplanes? Or can they be moved independently, or at different rates or to different positions (and what advantage would that offer)?

Finally, are there any aeroplanes on which they can be moved in the same direction?

On most general aviation airplanes and cable controlled aircraft they move together and for that matter are cross linked to the same physical cable. You can see this nicely on the control diagram for the B-24

(source)

Some ailerons are designed to counteract adverse yaw and may move such that they are asymmetrical to the flow over the wing.

1) Differential Ailerons: One aileron is raised a greater distance
than the other aileron is lowered. The extra upward aileron movement
produces more drag change than an increase in AOA on the downward
aileron. This produces an increase in drag on the descending wing,
which reduces adverse yaw.

2) Frise Ailerons: The aileron being raised pivots on an offset hinge.
The leading edge of the aileron is now pushed into the airflow,
creating drag and reducing adverse yaw. In this case, frise ailerons
are using form drag to counter induced drag.

But not all aircraft contain such systems. Fly-by-wire systems are theoretically capable of deflecting any control surface as they see fit and may not move symmetrically but generally speaking their direction of movement is opposite. Some fly-by-wire aircraft even have split ailerons which may see parts of a give aileron deflect differently. This answer covers that quite nicely.

If you really want to go back, the 1912 Farman HF.20 had single action ailerons that only deflected downward and were pushed back into position by the airflow. I would say (if these count as ailerons) that they do in fact not move in tandem. There were some other early designs covered here that worked a bit differently than modern ailerons.

As for your second point, aircraft that have elevons which serve as both the elevator and the aileron (mostly delta wing stuff like the Concorde and other high speed airframes) can move the surfaces in the same direction. Generally other surfaces are employed for the use case of needing to move both ailerons in the same direction (speed breaks).

The basic principle of ailerons is that the aileron on the down-going wing will deflect up, and the aileron on the up-going wing will deflect down. But the movement usually isn't equal on both sides.

Moving the aileron down increases the wings camber and therefore lift, which also comes with drag. This drag produces a yawing effect opposite to the desired turning direction. This is known as adverse aileron yaw, and needs to be overcome by coordinated use of the rudder when banking.

There are a few different ways of reducing this adverse yaw, the most common of which is known as differential ailerons. This is where the aileron that deflects up has a greater range of movement than the one that deflects down, reducing the drag on that side for the same banking force.

On a side note, airliners don't use ailerons at all at high speeds, because doing so will twist the wing. Spoilers are used instead for roll control in cruise as they are closer to the wing root, thus providing less twist. They operate only by deflecting up on the one side.