I do it all, facing the same way/biggest denomination at the back etc etc and have a system of gathering strict piles of the same total and they are kept in envelopes with amount and date saved etc written clearly on the front, this is for my own convenience and records. Office/cash desk background is to blame but I like it, it feels orderly and my life is like that too but not obsessively so.
One thing I don't like about money (the only thing) is the Scottish notes which have all kinds of different styles and characters on them, not much chance of keeping them all in a uniform way, but I don't get too many of them anyway.