Can a tapered BB be installed in either direction? I have a tapered bottom bracket on my bike, like the one in the pic. On this bike, as I bought it, the fixed cup is on the drive side of the bike, while the adjustable cup is on the non-drive side. In order to accommodate the derailleur mech, the fixed cup is not done up flush with the frame, and actually sticks out a few mm. If I adjust the BB so that the fixed cup is flush, the gears can no longer be made to work, so this extra length has to be in place on the drive side, but with this being the fixed cup with no locking ring, it's hard to get it to stay properly tightened.

My question is, is there a good reason the fixed cup is on the drive side, or is there nothing stopping me from swapping the cups around, having the fixed cup flush on the non-drive side and letting the adjustable cup offer up the extra distance on the drive side?
Short answer: no.

If you're lucky the axle is simply installed the wrong way round so rather than swapping the cups, swap the axle round and see if that fixes the problem.

The two cups have opposite threads, one left hand thread and one right hand thread. If you swap them and force them in you will cross thread the frame, and that's ugly to fix.

Unfortunately that also means you can't just get another lock ring and put it on the drive side as a spacer. But if you're lucky you'll be able to find another spacer from a modern BB with plastic spacers that fits. Or you could use a thread locking compound to reduce the amount of movement.

What you really should do is get a BB with a longer axle, or a derailleur that fits the bike. But those BB's are old or cheap, and if you're buying a new one you should get a sealed/cartridge type bottom bracket (of the correct axle length for your bike). That will fix the problem and give you a BB that doesn't need servicing.