Why are there inverted repeats in response elements?
Starting with prokaryotic (i.e., bacterial) transcription factors--which were the first to be characterized in depth, this type of pattern was noted.  Even for the "classic" Type II Restriction Endonucleases used for molecular cloning have palindromic recognition sequences.  In all three cases the answer is that the sequence-specific DNA binding protein binds as a dimer, and so each monomer is recognizing a "half-site" (one of the inverted sequences).