Are there different types of symmetry?
Symmetry is a very broad topic, and as was mentioned in the comments there are many kinds of symmetry. However, when dealing with molecules you should focus on point group symmetry. A good place to start is this link.

But let's get back to the question:

You mention that you know water is asymmetrical and is therefore polar but at the same time you also note that you CAN draw a line through it to cut it into two equal halves. In that way, you can say water has a mirror plane and is therefore symmetric. This type of symmetry is more precisely defined as $C_{2V}$. Ammonia has  $C_{3V}$, Benzene has  $D_{6h}$, and so on and so forth. 

Now, symmetry and polarity do have some relation but you can't tie them together, the case in point being the hydrocarbon $\ce{CHMeEtPr}$. From the point of view of symmetry, this is definitely not a symmetric molecule. And despite its complete asymmetry this is definitely what we would call a non-polar molecule.

So are there any properties related to symmetry? Yes - the ability for a molecule to rotate plane polarized light. We measure this and call it optical activity, and call the asymmetric molecules chiral.

Hope that helps!