I am about to hook an adjustable U3V70A of these up to a 6V battery and need 9-12V output (precise value unknown, and for a problem I have no control over). How can I make sure that I am set to 20V so the output is not lower than the input? Adjust the pot fully clockwise or fully counterclockwise? If I set it to 20V I can back it down; if I set it to 4V and put 6V in you say I am at risk of damaging it.
Turning the adjustment screw clockwise increases the output voltage, and turning it counterclockwise reduces the output voltage. So, one option is to turn the adjustment screws clockwise several times before you power the regulator.
A better option though is to temporarily use a lower supply voltage (e.g. around 3-4V, which you could achieve with a few AA batteries) when you first power the regulator. I also suggest having no load connected to it until after you have adjusted the output to where you want it.
By the way, the adjustment screws for these potentiometers do not have physical hard stops. There is no danger of damaging the pot by turning the screw too far, but the output voltage setting will just not adjust any higher/lower depending on the direction you are turning it. Also, if you overshoot the limit, you will need to backtrack before it starts adjusting again.