Ah yes, good ol' output inversion.
Basically, pulling the input low, causes the input stage to shut down. No current gets to the output stage, which makes it think it should deliver +MAXVAL output...
Didn't know that a series resistor can "fix" that on TL081, interesting. I guess what's happening is, the input stage's current gets shunted through the input pin, and with the large series resistor, not enough current can be shunted to make it screw up.
This makes more sense if you look at the internal diagram of the amp (if you can find it -- a lot of datasheets don't bother showing
) and think about diode paths from the input pin. IIRC, they used P-channel JFETs. A JFET is a fancy diode, a P-ch being the channel (drain to source) as the anode, and the gate as the cathode. Pull the gate low, the diode turns on, and the channel-to-gate diode must pull the channel down to ~0.6V above Vin.
P-ch FETs have the drain terminals on the "bottom" (i.e., the negative side), which carry the signal to the rest of the amp. Probably, it's arranged so that when one JFET's current drops, the output rises (i.e., there is an intermediate gain stage, such that, when less current goes into its input, it makes a higher voltage at its output). So, pulling that current away, forcibly, causes the output to rise, to +MAXVAL.
Which also means you should be able to measure the input stage's bias current this way: by seeing how much current comes out of an input pin when pulled low.
IIRC, the trim pins control input bias currents as well, so you could play with that too, I suppose.
But yeah, this won't make any sense if you didn't find a diagram, or probably even if you did...
Tim