On a single piece of silicon on which there is more then one op amp, are the input offset voltages at all correlated, i.e. would they be expected to be same direction and similar magnitude?
No, you can't assume anything about correlation between opamps on the same chip unless the datasheet explicitly says so. I don't remember ever seeing a datasheet say anything about offset voltages of opamps on a chip relative to each other.
Think about it: offset voltages are due to the slight mismatches between transistors in a chip. Some transistor parameters are random, but others may correlate with where they are on the wafer. However, the input transistors of an opamp are already near each other, and likely much closer to each other than the input transistors of other opamps on the same chip. This doesn't leave much mechanism for the offset voltages between nearby opamps to be correlated somehow.
The input transistors in this TL072 are interleaved so that they have the same center, so that if the transistor parameters are varying linearly across the die, the average parameters are the same. Despite this the transistors are still slightly different because the variation is not exactly linear.
So if the two transistors right next to each other are mismatched, why would the ones on the other half of the die have the same mismatch?
For absolute values-NO. For drift over time, they will have there independent offsets and gain tracking issues as well.
For dual and quad op-amps, precision means independent gain and offset for each channel.
With independent gain and offset, there should be some correlation over time, but no datasheet would ever state that. Reason is that the user ambient temperature and voltage and loading of outputs is unknown.
If one of four channels has a heavy load, any hint of correlation is gone.