They do turn opposite or "spin" to some degree.
For the most part, U.S. aircraft are built with propellers that turn clockwise (right) when viewed from the cockpit. Because of this, they have left turning tendencies. The left turn tendency is caused by three things:
- P Factor: the downward blade (on the right side) has a higher angle of attack so it gets a "bigger bite" of the air. This causes a left yaw.
- Slip Stream: The the air from the prop goes in a spiral around the aircraft and ends up hitting the left side of the vertical stabilizer, again causing a left yaw.
- Torque: The prop turns to the right so the engine tries to go opposite (to the left). Since the engine is attached to the aircraft, the aircraft tries to roll left.
There is a fourth turning tendency, but it can be in any direction. During a climb it is a right turn tendency. It is called gyroscopic precession. A spinning prop has some of the properties of a gyroscope. If you apply a force to a gyroscope, you will get a resultant force (precession) occurring at a point that is 90 degrees in the direction of the spin. When you climb, you apply a force to the top of the propeller (i.e. push it back so it angles upward). The resultant force is felt 90 degrees to the right where the prop will also be pushed back, causing a right yaw. It is just opposite in a descent (left yaw). Similarly a left turn or yaw will cause a upward precession and a right turn or yaw a downward precession.
The aircraft are engineered so that in level flight at cruise speed, these forces are "trimmed" out. This is usually done with trim tabs, either fixed or movable. The size of the vertical stabilizer and rudder have a lot to do with this too. During descents, when there is less power on the engine, these factors are less noticeable. During takeoff and climb, when there is more power producing more of these forces, the pilot uses right rudder to counteract them.
I hope this helps but I'm not an aero guy and this is just my layman's explanation. Anything more just gives me headaches.