ける(かたづける) To tidy up, to straighten, to take a messy pile of crap on your desk, and make it neat.   $$$
(せいり) To organize – books by author, say, or CDs by genre.
(はいれつ) To put experimental data in a sequence. A very technical, limited word.
える(ととのえる) To arrange or prepare – usually used with “prepare a hairstyle”, or “clean up a sentence on a re-write of a paper”, and not used anywhere else. Sometimes books will refer to someone who is pretty as having a 整った顔  (literally:a well-arranged face!)