Exactly half of this idea is doable:
one half of the crust & core rotating against the planetary rotation
As the other answer explains, having the entire half of the planet rotating the wrong way would immediately grind to a halt due to the friction between the two halves. So we can't have the core part.
But the crust part - that would be doable if you were to give up on hard science. The crust has relatively little mass compared to the magma and all the other stuff above the core. With enough handwavium (but certainly less than you'd need to make two counter-rotating halves work) you could make different parts of the planetary crust rotate in different directions, or at least at different rates.
This becomes a lot easier if the substrate is more malleable. For example, if your tectonic plates were more like sheets floating on water or air, rather than thick rock slabs floating in magma. Your mental model should be more like a gas giant than a rocky planet, where counter-rotating matter is par for the course.
And of course this all becomes much easier with a planet that rotates slowly. Earth is a spry young thing zipping along at 1 rotation every 24 hours. If you were looking at 1 rotation every year (a tidally locked planet) or even less frequently, you could get away with a lot more silly stuff.
At this point we are so far in the realm of contrivance that you could have whatever prevalent conditions you wanted on the surface. This is not a hard-science premise.