So people who start to drive cars notice something interesting about the brake pedal: it “pitches you forward.” And this makes your passengers a little bit sick if you don't do it carefully. The explanation is pretty simple in the rest frame of the ground: your body has momentum, momentum is conserved and inertia is a thing so your body wants to keep moving, the car slows down, so your body slides forward until the seat belt can pull it back and remove that forward momentum. This gets a little harder to think about in the reference frame of the car, where of course the body has no forward momentum. A high school teacher who does not want to engage with the problem might just tell you that it's wrong to look in that frame because that frame is not inertial. One can alleviate this by looking at the “starting frame” of the car before it undergoes deceleration, which of course makes it go “backwards” in this frame. Instead of saying that the brakes stop the car, you would kind of say that the brakes try to stop the road. The car gets dragged backwards simply as a result of a third-law force pair.
But, there is an alternative description in terms of angular momentum, which is also conserved. And this is a little more interesting maybe. In the same car’s starting reference frame, there's some little angular momentum held in the wheels, but the car itself doesn't have any because it's not moving in that frame of reference. And the wheels can be assumed to basically be massless compared to the tonnage of the car. Let's say “pitching forward” about the front wheel axle is the “positive” direction of angular momentum, the car is not pitching forward before the brake is engaged, basically zero angular momentum.
But, and here's the main point, the road moving backwards has all of this positive angular momentum because it's underneath the axle: two negatives make a positive. Before the brake is engaged, the wheels are able to move pretty freely and so the road is not able to share any of this angular momentum with the car. This sometimes comes about when we talk about driving on a conveyor belt (or airplane on a conveyor belt), if you were to hire a mythical titan/giant being to pick up the road and yank it backwards, if the wheels are massless and frictionless then the car won't feel any force from this.
And what does the brake do? It couples the reservoirs. Friction allows two separate systems to share their angular momentum. The road’s positive angular momentum floods into the car and pitches the whole car forward, you get pitched because you're attached to the car and the car is pitched forward. What resists this torque, why doesn't the car just flip end-over-end? The weight of the car itself! The center of mass of the car has to be a certain distance behind that front axle, so that as the weight transfers from being equally loaded across both axles to being dominantly on the front one, gravity torques in the opposite direction. And, see, that's a new thought: when we were looking just in terms of momentum it might have never occurred to you that having this heavy engine block in front of the axle actually makes this braking motion more nauseating.
In the planetary case, one wants to distinguish two different sources of angular momentum: spin and orbital. The Earth rotates about its axis relative to the distant Sun, once a day. The Moon orbits about the Earth roughly once a month. The moon also spins about its axis relative to the distant Sun, also roughly once a month, so that the face we see is always approximately the same.
Note that this has a very different explanation from the rotation (relative to the sun) of my car once per day. My car has strong friction holding it to the Earth: the friction shares angular momentum between the Earth and spin and orbit until they all reach a shared state of rotational equilibrium where everything behaves roughly like one big rigid body. The Moon doesn't have that though. The Moon is in what we call a “1:1 resonance” between spin and orbit, even though it has no friction with Earth and indeed the times don't line up: 1 month vs 1 day. While the explanation does involve tidal forces (it is called “tidal locking,” if you want to search it), 1:1 is only one possible option for how these balance out and some things are in, say, a 3:2 resonance instead. The point is that the tidal force is subtly coupling the spin and the orbit without coupling either to the spin of Earth.
One interpretation of your asteroid example, is to think of the International Space Station as precisely this sort of asteroid, it goes about the Earth something like two times per day, it has very little friction with Earth’s atmosphere, it is just constantly falling perpendicular to Earth. Because it doesn't get close enough to have that friction and share its orbital angular momentum with Earth’s surface. Like the moon it is in a 1:1 resonance (I think?) But that's because we nudged it there that way with small adjustment rockets. If the asteroid had a velocity greater than the ISS but approached as close, either it would be captured into an elliptical orbit or else it would be sent off on its way along a hyperbola. We use these hyperbolas all the time too for our spacecraft; if you can arrange the hyperbola “behind” the planet in terms of its motion about the sun, the planet will essentially “pull up” the gravitational potential “out from underneath” the spacecraft and it will emerge from the approach with a lot of extra speed: this is called a “gravitational slingshot” maneuver.
