Why don't NK-12 powered contra-rotating propellers turn in equal rpm?
The NK-12 engine uses a differential epicyclic (planetary) gearbox. Unlike the typical planetary gearboxes where one of the gears is fixed, here all three parts are moving: the turbine drives the sun, the planet pinion drives the front propeller, and the ring drives the rear propeller.

The gear ratios are the same between the front and rear, but the torque is not. The difference you see has to do with this torque difference, as well as with different rotating masses and friction during start. The torque is what is actually important.

If you hold one part, then the other two will be in the expected sync. Say, if you jam one prop and start the turbine, it will simply spin the other prop twice as fast. (But if you jam the turbine, things will be less intuitive: the props should turn in the same direction if you move one by hand). Now if you allow the turbine to free spin, you'll have two degrees of freedom, and the eventual relative RPM in such system will depend on the torque on each prop.

In the explanation below, I use this work by S. Falaleyev et al (pdf in Russian). Here is the diagram of the gearbox:



If we assume the torque balance such that the RPMs are the same (except the sign), the gear ratio will be (I skip the derivation):

$$i = 1 + 2(\frac{z_2 \cdot z_4}{z_1 \cdot z_3})$$

($z$ are the teeth numbers according to the diagram). For the given numbers we get $11.33$ (or rather, it should be $1/11.33 = 0.088$ reduction). The nominal speed of the props was just 735 RPM!

At the same time, the torque ratio (front to rear) is

$$\frac{M_f}{M_r} = -(1 + \frac{z_1 \cdot z_3}{z_2 \cdot z_4}) \approx -1.19$$

Thus, the front prop gets almost 20% more torque than the rear prop. This is why it spins up faster on start-up. (It is also lighter than the rear one).

At the nominal speed, the governor automatically sets a finer blade pitch on the rear prop to equalise the speeds. (Why not just accept higher speed/thrust from the front prop? Apparently, because the prop is limited by the tips speed, and we don't want the front one go supersonic while the rear one is relaxed).