Yes. The more so the higher the inertia and speed of the rotating masses and the lower the control authority. While the gyro forces of a jet are negligible in regular flight due to the high damping provided by the wings and tail, in hover they can become dominant. The Bristol-Siddeley Pegasus engine of the Kestrel, Harrier and AV-8B jump jets needs to have the low pressure spool run in opposite direction of the high pressure spool to balance its gyro effects. If that would not be the case, a yawing motion would produce a pitching motion, and vice versa. When you sit on a jet, the thrust of which is equivalent to your weight, tilting this jet fore or aft only slightly will produce a quick shift of your location, so any maneuvering in hover will become extremely hard.
USMC AV-8B Harrier in hover (picture source, © Richard Seaman 2001)
Generally, contra-rotation of engine parts has been used to reduce gyro effects. They first became an issue with rotary engines in WW I. A rotary engine has its crankshaft fixed to the airplane, and both the cylinder block and the propeller rotate. This gives better cooling at low speed and produces a flywheel effect, so the engine runs more smoothly. But when you yaw, the gyro effect pitches the aircraft up or down, so any precise maneuvering becomes very hard.
With the increasing engine power in 1916 and 1917, this effect became so severe that geared engines were developed where the cylinders rotate in one direction and the propeller in the opposite direction. As a consequence, the propeller had only half the RPM in air as it had with the cylinder block. This resulted in great propeller efficiency, but also big propeller diameters, so airplanes with those engines needed a high landing gear. Below is a picture of a Roland D XVI with a Siemens & Halske III contra-rotating rotary engine from 1918. This was an excellent fighter aircraft for its time with almost no gyro coupling.
Roland D XVI (picture source)
Generally, jet engines are sensitive to maneuvering: Changing load factors will bend the spools between their bearings such that the tip clearance of both the turbine and the compressor need to be big enough to avoid contact under g forces, or when rolling with a wing-mounted engine. Every engine manufacturer will publish a load factor envelope within which the engine is safe to operate. Widening this envelope will reduce engine efficiency because the gaps between the rotating parts and the engine housing need to grow with the maximum load factor.