Is motion really relative? I had a question that had been bothering me for some time. I tried to come up with an explanation but I wanted to know whether it is right. 

Imagine two different observers say A & B floating in space in their corresponding frames say P&Q with nothing else surrounding them. A observes that B is accelerating away from him and B observes that A is accelerating away from him. I feel that this situation is completely symmetrical for both A & B and thus if the motion is actually relative, then the observations of both A & B must be similar. However, then how is it possible that only one of them say, A, happens to observe a ball in his hand following Newton's laws of motion showing that he is in an inertial frame of reference whereas a ball in the hands of B seems to be acted upon by pseudoforces? 
  Basically, my question is what is the reason for the seemingly asymmetry between them ?

 A: Motion is relative but acceleration is absolute. You can know if you are being accelerated without any reference to the outside world. You will feel it in your internal organs and if you carry out an experiment you will observe pseudo forces, even if you don't look outside your space ship.
If a body A feels none of these effects and sees that another body B is being accelerated ´relative´ to it, then A is in an inertial frame (but it is unknown whether it is at rest, since this is relative) and B must be accelerating. This is not symmetrical, since B will feel an acceleration.
A: 
...what is the reason for the seemingly asymmetry between them ?

People and balls and most other things don't just accelerate. If you are accelerating, it's because something is pushing you. If only one of "A" or "B" is feeling the push, then that obviously is not a "completely symmetrical" situation.

how is it possible that...a ball in the hands of B seems to be acted upon by pseudoforces?

If B thinks that a pseudoforce is acting on the ball, that's because a real force is acting on B, but not acting on the ball.  B chooses to ignore the force that they feel, and pretends that their own frame of reference is unaccelerated.
A: There are really 3 cases possible in this situation:
1) Only A accelerates with respect to an inertial frame,
2) Only B accelerates with respect to an inertial frame,
3) Both of them accelerate 
In all of the cases mentioned above we can get the same relative acceleration, and this is why we will not be able to decide which case is actually happening if the only information that we have is about their relative acceleration.
In a sense, the situation is not symmetric at all. But, it becomes symmetric when we look at it from a reference frame P or Q. This is similar to how we cannot differentiate between us accelerating towards the Earth and a lift in space accelerating towards us with an acceleration equal to $g$.
