[spirochete]

Why does my book say that hydrogen peroxide only has a C2 (180 degree rotation) symmetry element. 

This means they are giving it the lowest possible conformation for symmetry.  In a higher symmetry conformation it could have two mirror planes and at least one more rotational axis.  This is the opposite of how I learned to identify symmetry elements in organic chemistry. 

What gives?

[LQ43]

Its been a long time since I did this (so could be way off) but look at the molecule in 3D - the lone pairs on the oxygens hinder free rotation. In a Fischer projection, the hydrogens are staggered. The eclipse form may be too thermodynamically unstable.

[FeLiXe]

yes, the equilibrium conformation of H2O2 is not planar

[spirochete]

Interesting.  From steric arguments you'd expect a full anti conformation.  I wonder if the deviation from this is caused by the favorable interaction between the oxygen lone pair and the O-H antibonding orbital.

[FeLiXe]

i always took it for granted ...
but i guess it has to be an electronic reason