Jim,
Your Tascam 22-4 isn't necessarily meant to hook up in surround sound like that. You're thinking along the lines of a "quad" machine (The Quadraphonic system from the early-mid '70s), in which case it would wire up in a configuration simliar to what you've mentioned.
If you want to play 4-track stereo tapes on your 22-4, you'll have to hook up Channel 1 for Left, and Channel 3 for Right. If you listened to a regular stereo tape through tracks 2 and 4, you'd be hearing the other side of the tape backwards! Four-track recorders of the format you have are meant for professional recording where a multitrack mixer is fed into 4 recording buses, which are recorded on the 4 discrete channels on the R2R in one winding direction. Regular stereo tapes use tracks 1 and 3 for a stereo pair, so when the end of the tape is reached, you'd flip it over and use "tracks 1 and 3 on the other side," which correspond to 2 and 4 on "Side A" (the direction that is played when you first thread the tape).
So, if you have discrete 4-track tapes (all 4 tracks recorded in 1 direction), you'll have to somehow feed the 4 outputs from the R2R into a mixer where you can pan them to L and R accordingly. But if you're only playing stereo tapes, run channel 1 to left and 3 to right as I mentioned above.
Auto-reverse R2Rs play tracks 1 and 3 for Side A, then 2 and 4 for Side B with the winding direction reversed.
Hopefully this helps. I have a Tascam 34B (similar to your deck but 10.5" reels) and I'm looking for a 4-bus mixer to use with it so I can take advantage of the design.