I have 20 amp circuit with 12/3 wire and want to attach 10/3 wire to it. Can't this be done?

Yes, but WHY? Just use the right wire.

As long as the wire has equal or greater capacity than the breaker, no problem.

Creative -- (?)

Would you like to re-phrase that response?

10 AWG conductors are rated for 30 amps max.

12 AWG conductors are rated for 20 amps max.

Yes you may join 10 AWG to 12 AWG, but your breaker size would be limited to 20 Amps.

You can - but some time from now "somebody" might get the wrong idea about the intent at the 10/3 end. Make sure it is DOCUMENTED.

Only reason to oversize wires (in my mind) would be to prevent a long run/voltage drop scenario - say, a table saw "way out there in the garage...."

Not a good idea. Somebody may see the 10/3 and ASSUME it's a 30 amp circuit, and install a breaker accordingly. Just looking for trouble using 2 different wire sizes.