The correct answer is $\boxed{\text{C}}$.
In a dynamometer 3-phase power factor meter, the planes of the two moving coils are at $90^\circ$ to each other. This is because the two coils are connected to different phases of the power supply, and the voltage and current in each phase are 90 degrees out of phase with each other. This means that the torque on each coil is at a maximum when the planes of the coils are at $90^\circ$ to each other.
If the planes of the coils were at $0^\circ$ to each other, the torque on each coil would be zero, and the meter would not register any power. If the planes of the coils were at $60^\circ$ or $120^\circ$ to each other, the torque on each coil would be at a maximum, but the meter would not be able to distinguish between the two phases.
Therefore, the planes of the two moving coils in a dynamometer 3-phase power factor meter must be at $90^\circ$ to each other in order for the meter to work properly.