chosing a cassette deck

I have a 1980s vintage Yamaha with auto reverse as you describe. It plays each side and stops. I don't recall the model number, and the unit is boxed and stored at the moment. It is a single deck, though. If you are interested in the model number, I can dig it up.

The best auto-rev. decks are single cassette units. The Nakamichi Dragon is considered the absolute best of these -- primarily because it has automatic head azimuth adjustment. Since the head needs to jump up or down slightly, depending which direction the tape is going, most auto-rev decks don't give the same reproduction as a fixed-head single-direction deck. A properly-adjusted Dragon will however. They're available on Ebay for anywhere from $350 to $1250 depending on condition and accessories (a wired remote is available)

I had a single deck JVC that had a selector allowing you to play one side, both, or continuous. (Just what you want i think.) I'm not sure if their duals had this but it was a nice player. I still have it and use it once and a while.

Try a Nakamichi. You wont' be dissapointed. If you need to dub, buy two decks. Auto-reverse was only available on the Dragon I think. There were a few models that actually flippped the tape. The reason behind this was for optimal tape alignment in one direction only.