Thanks for the photo. That is an ornamental rim lock made by the Russel & Erwin Company and the REO inside a large C was their trademark. The lock was made at the end of the 1800’s or beginning of the 1900’s. When originally sold there was also a strike or keeper with a similar and complimentary design that could be monuted on the jamb to receive the bolt.
It should be an easy matter for a locksmith to fit a key since the cover can be removed to see the internal mechanism which could be simple wards or there could be a lever or two. I believe the knobs sold with the lock were also made with a similar design. You are correct that the lock could be mounted for either a left or right hand door, but more importantly could be accessed by the key from either side of the door by simply making a hole through the door. There would also have been an escutcheon for the opposite side of the door with a complimentary design upon it.
The number may have been the actual part number it was sold under and the 2 is very likely the key number that should operate it. Had it been a padlock it would fall under the general description of ‘story lock’ as there are many padlocks with similar designs. There is a link from this site to a site that is a great reference for story locks and while I have not noticed it, they may have a section that includes locks similar to yours. Search here for story lock to find the link. Hope that helps.
BBE.
