Having done it if you put led bulbs in flashers and the single idiot light all four flashers work whichever way you put the switch because the idiot light connects both sides together, the resistance of the ordinary filament bulbs is what lets the system work
I didn't believe this at first, owning only a K8 (with 2 idiot flasher lights), but looking at the wiring diagram for the earlier models, I'm shocked to see you're right.
First and foremost: A well deserved criticism for the engineers that did this -- theres absolutely no reason it couldn't operate like the buzzer mentioned in the diagram (never seen a turn signal buzzer in my region), where the dash light is BEFORE the switch, only operating when the signals are engaged. This is engineers pulling fancy tricks because they're bored. I'm a bit hypocritical here, I do it myself in software, but it's overengineered for kicks, not for any real good reason.
Second: Unless I'm missing something, diodes won't help. Well they will, but only in that the indicators will work, but I don't see how you can use diodes to get the idiot light to work in both directions while also fixing the indicators themselves. ON SECOND THOUGHT: Idiot light blue to ground, idiot light orange to both orange AND blue with diodes on them should do the trick.