This is probably a personality-temperament thing, but I have never felt remotely awkward going to the cinema alone. I simply could not give twin defecations what anyone thinks about me or my filmgoing habits. I have been to the cinema by myself as soon as I was old enough to do so (around the age of 12) and have continued that practice to this day. It is a greatly underrated pleasure, and whilst I do also enjoy going with my wife and children, I'm never bothered if I end up going alone.
More of my thoughts on the subject here, if you're interested. :)