In my textbook it was written unpolarized light was symmetric about the direction of propagation of light(EM wave)but not so in plane polarized light ....can someone please explain to me as how this is?
Unpolarized light is statistically symmetric about the direction of propagation, because every photon has an equal probably of having its $E$-field in some direction normal to the direction of propagation. Therefore, if you put a polarizer in its path, no matter how you turn it, it will always let the same intensity through.
For linearly polarized light, however, all photons have their $E$-field in the same plane, therefore the intensity changes when you turn the polarizer, i.e. it is not symmetric anymore.